1st PUC Maths Question Bank Chapter 13 Limits and Derivatives

   

Students can Download Maths Chapter 13 Limits and Derivatives Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 1st PUC Maths Question Bank Chapter 13 Limits and Derivatives

Question 1.
Explain the meaning of x → a .
Answer:
Let x be a variable and ‘a’ be a constant. Since ‘x’ is a variable we can change its value at pleasure. It can be changed so that its value comes nearer and nearer to a. Then we say that x approaches ‘a’ and it is denoted by x → a .

KSEEB Solutions

Question 2.
Investigate the behaviour of \(f(x)=\frac{x^{2}-4}{x-2}\) at the point the point x = 2 and near the point x = 2.
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 1
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 2
It is clear from the table that as gets nearer and nearer to 2 from either side, f(x) gets closer and closer 4 from either side.
\(\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=4\)

Question 3.
Define limit of a function.
Answer:
Let f(x) be a function defined on an interval that contains x = a, except possible at x = a. Then we say that, \(\lim _{x \rightarrow a} f(x)=L\)
If for every number ∈ > 0 there is some number δ > 0 such that
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 3

Remark: We say  \(\lim _{x \rightarrow a^{-}} f(x)\)is expected value of f at
x = a given the values of f near x to the left to a.The values is called left hand limit.
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 4
\(\text { We say } \lim _{x \rightarrow a^{+}} f(x)\) is expected value of f at x = a given the values of f near x to the right to a. The value is called right hand limit.
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 5

Question 4.
Discuss the limits of the function
\(f(x)=\left\{\begin{array}{ll}{-1,} & {\text { if } x<0} \\{1,} & {\text { if } x>0}\end{array} \text { at } x=0\right.\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 6
∴ LHL ≠ RHL
Limit does not exist

Question 5.
Discuss the limit of the function f(x) = x +10 at x = 5.
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 7

Question 6.
Discuss the limit of f(x) = x3 at x = 1.
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 8

Question 7.
Find \(\lim _{x \rightarrow 2} f(x), \text { where } f(x)=3x\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 9

Question 8.
Find \(\lim _{x \rightarrow 2} f(x)\),where f(x) = 3 a constant function
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 10

Question 9.
Find \(\lim _{x \rightarrow 1}\left(x^{2}+x\right)\)
Answer:
\(\lim _{x \rightarrow 1}\left(x^{2}+x\right)=1^{2}+1=2\)

KSEEB Solutions

Question 10.
Find
\(f(x)=\left\{\begin{aligned}x-2, & x<0 \\0, & x=0 \\x+2, & x>0\end{aligned}\right.\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 11
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 12

Question 11.
Find \(\lim _{x \rightarrow 1}\left(x^{3}-x^{2}+1\right)\)
Answer:
\(\lim _{x \rightarrow 1}\left(x^{3}-x^{2}+1\right)=1^{3}-1^{2}+1=1\)

Question 12.
Find \(\lim _{x \rightarrow 3} x(x+1)\)
Answer:
\(\lim _{x \rightarrow 3} x(x+1)=3(3+1)=12\)

Question 13.
Find \(\lim _{x \rightarrow-1}\left(1+x+x^{2}+\ldots+x^{10}\right)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 13

Question 14.
Find \(\begin{aligned} &\lim (x+3)\&x \rightarrow 3\end{aligned}\)
Answer:
\(\lim _{x \rightarrow 3}(x+3)=3+3=6\)

Question 15.
Find
\(\lim _{x \rightarrow \pi}\left(x-\cfrac{22}{7}\right)\)
Answer:
\(\lim _{x \rightarrow \pi}\left(x-\cfrac{22}{7}\right)=\pi-\cfrac{22}{7}\)

Question 16.
Find
\(\lim _{x \rightarrow 1} \pi r^{2}\)
Answer:
\(\lim _{x \rightarrow 1} \pi r^{2}=\pi(1)^{2}=\pi\)

KSEEB Solutions

Question 17.
Find
\(\lim _{x \rightarrow 4} \cfrac{4 x+3}{x-2}\)
Answer:
\(\lim _{x \rightarrow 4} \cfrac{4 x+3}{x-2}=\cfrac{4(4)+3}{4-2}=\cfrac{19}{2}\)

Question 18.
Find
\(\lim _{x \rightarrow-1} \cfrac{x^{10}+x^{5}+1}{x-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 14

Question 19.
Find
\(\lim _{x \rightarrow 0} \cfrac{(x+1)^{2}-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 15

Question 20.
Find
\(\lim _{x \rightarrow 2} \cfrac{3 x^{2}-x-10}{x^{2}-4}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 16

Question 21.
Find
\(\lim _{x \rightarrow 3} \cfrac{x^{4}-81}{2 x^{2}-5 x-3}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 17

Question 22.
Find
\(\lim _{x \rightarrow 0} \cfrac{a x+b}{c x+1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 18

Question 23.
Find
\(\lim _{x \rightarrow 1} \cfrac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c \neq 0\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 19

Question 24.
Find
\(\lim _{x \rightarrow 2} \cfrac{\cfrac{1}{x}+\cfrac{1}{2}}{x+2}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 20

KSEEB Solutions

Question 25.
Find
\(\lim _{x \rightarrow 1} \cfrac{x^{2}+1}{x+100}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 21

Question 26.
Find
\(\lim _{x \rightarrow 2} \cfrac{x^{3}-4 x^{2}+4 x}{x^{2}-4}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 22
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 23

Question 27.
Find
\(\lim _{x \rightarrow 2} \cfrac{x^{2}-4}{x^{3}-4 x^{2}+4 x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 24

Question 28.
Find
\(\lim _{x \rightarrow 2} \cfrac{x^{3}-2 x^{2}}{x^{2}-5 x+6}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 25

Question 29.
Find
\(\lim _{x \rightarrow 1}\left[\cfrac{x-2}{x^{2}-x}-\cfrac{1}{x^{3}-3 x^{2}+2 x}\right]\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 26

Question 30.
Prove that,for any positive integer n,
\(\lim _{x \rightarrow a} \cfrac{x^{n}-a^{n}}{x-a}=n\left(a^{n-1}\right)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 27

Question 31.
Find
\(\lim _{x \rightarrow a} \cfrac{x^{3}-a^{3}}{x-a}\)
Answer:
\(\lim _{x \rightarrow a} \cfrac{x^{3}-a^{3}}{x-a}=3 a^{2}\)

KSEEB Solutions

Question 32.
Find
\(\lim _{x \rightarrow 2} \cfrac{x^{7}-128}{x-2}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 28

Question 33.
Find
\(\lim _{x \rightarrow 1} \cfrac{x^{3}-1}{x-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 29

Question 34.
Find
\(\lim _{x \rightarrow a} \cfrac{x^{2 / 7}-a^{2 / 7}}{x-a}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 30

Question 35.
Find
\(\lim _{x \rightarrow-a} \cfrac{x^{5}+a^{5}}{x+a}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 31

Question 36.
Find
\(\lim _{x \rightarrow-1} \cfrac{x^{3}+1}{x+1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 61

Question 37.
Find
\(\lim _{x \rightarrow a} \cfrac{x \sqrt{x}-a \sqrt{a}}{x-a}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 32

Question 38.
Find
\(\lim _{2 x \rightarrow-1} \cfrac{8 x^{3}+1}{2 x+1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 33

Question 39.
Find
\(\lim _{x \rightarrow 1} \cfrac{x^{15}-1}{x^{10}-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 34

Question 40.
Find
\(\lim _{x \rightarrow 2} \cfrac{x^{3}-8}{x^{2}-4}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 35

KSEEB Solutions

Question 41.
Find
\(\lim _{x \rightarrow 4} \cfrac{x^{3}-64}{x^{2}-16}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 36

Question 42.
Find
\(\lim _{x \rightarrow 2} \cfrac{x^{10}-1024}{x^{5}-32}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 37

Question 43.
Find
\(\lim _{x \rightarrow 9} \cfrac{x^{\cfrac{3}{2}}-27}{x-9}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 38

Question 44.
Find
\(\lim _{x \rightarrow a} \cfrac{x^{\cfrac{2}{3}}-a^{\cfrac{2}{3}}}{x^{\cfrac{3}{4}}-a^{\cfrac{3}{4}}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 39

Question 45.
\(\lim _{x \rightarrow 0} \cfrac{\sqrt{1+x}-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 40

KSEEB Solutions

Question 46.
Find
\(\lim _{z \rightarrow 1} \cfrac{z^{\cfrac{1}{3}}-1}{z^{\cfrac{1}{6}}-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 41

Question 47.
Find
\(\lim _{x \rightarrow 0} \cfrac{(1-x)^{n}-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 42

Question 48.
Find
\(\lim _{x \rightarrow 1} \cfrac{\left(x+x^{2}+x^{3}+\ldots+x^{n}\right)-n}{x-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 43

Result 1:
Let f and g be two real valued functions with the same domain such that f(x) ≤ g(x), for all x in the domain of definition. For some a, if both
\(\lim _{x \rightarrow a} f(x)\)and \(\lim _{x \rightarrow a} g(x)\) exists them. \(\lim _{x \rightarrow a} f(x) \leq \lim _{x \rightarrow a} g(x)\)

KSEEB Solutions

Result 2 : (Sandwich theorem):
Let f, g and h be real functions such that f(x)≤g(x)≤h(x),∀x, in common domain of definition. For some real number a
\(\lim _{x \rightarrow a} f(x)=l=\lim _{x \rightarrow a} h(x) \text { then } \lim _{x \rightarrow a} g(x)=1\)

Question 49.
Prove that
\(\lim _{x \rightarrow 0} \cfrac{\sin x}{x}=1\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 44
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 45
Question 50.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\sin a x}{b x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 46

KSEEB Solutions

Question 51.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\sin a x}{\sin b x}, a, b \neq 0\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 47

Question 52.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\sin 4 x}{\sin 6 x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 48

Question 53.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\tan x}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 49

Question 54.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\tan 3 x}{\sin 2 x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 50

Question 55.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\tan 8 x}{\sin 2 x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 51

Question 56.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\sin 5 x}{\tan 3 x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 52

KSEEB Solutions

Question 57.
Evaluate:
\(\lim _{x \rightarrow \pi} \cfrac{\sin (\pi-x)}{\pi(\pi-x)}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 53

1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 62

Question 58.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\cos x}{\pi-x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 54

Question 59.
Evaluate:
\(\lim _{\theta \rightarrow 0} \cfrac{1-\cos 4 \theta}{1-\cos 6 \theta}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 55

Question 60.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{1-\cos 5 x}{1-\cos 6 x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 56

KSEEB Solutions

Question 61.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{1-\cos 3 x}{x^{2}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 57

Question 62.
Evaluate:
\(\lim _{\theta \rightarrow 0} \cfrac{1-\cos \theta}{2 \theta^{2}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 58

Question 63.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{1-\cos x}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 59

Question 64.
Evaluate:
\(\begin{aligned}&\lim x \cdot \sec x\\&x \rightarrow 0\end{aligned}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 60

KSEEB Solutions

Question 65.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{a x+x \cos x}{b \sin x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 62

Question 66.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\sin a x+b x}{a x+\sin b x}, a, b, a+b \neq 0\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 66

Question 67.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{1-\cos x}{\sin ^{2} x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 67
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 68

Question 68.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{1-\cos 2 x}{3 \tan ^{2} x}\)
Answer:

1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 69

Question 69.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\tan 2 x+\sin 2 x}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 70

Question 70.
Evaluate:
\(\lim _{x \rightarrow 0}[\csc x-\cot x]\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 71

Question 71.
Evaluate:
\(\lim _{x \rightarrow \cfrac{\pi}{2}} \cfrac{\tan 2 x}{x-\cfrac{\pi}{2}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 72

KSEEB Solutions

Question 72.
\(\begin{aligned} &\text { Find } \lim _{x \rightarrow 0} f(x) \text { and } \lim _{x \rightarrow 1} f(x) \text { if }\\&f(x)=\left\{\begin{array}{ll}{2 x+3,} & {x \leq 0} \\{3(x+1),} & {x>0}\end{array}\right.\end{aligned}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 73

Question 73.
Find
\(\lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\{\begin{array}{cc}{x^{2}-1,} & {x \leq 1} \\{-x^{2}-1,} & {x>1}\end{array}\right.\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 74

Question 74.
Evaluate
\(\lim _{x \rightarrow 0} f(x), \text { where } f(x)=\left\{\begin{array}{cc}{\frac{|x|}{x},} & {x \neq 0} \\{0,} & {x=0}\end{array}\right.\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 75

Question 75.
Find
\(\lim _{x \rightarrow 5} f(x), \text { where } f(x)=|x|-5\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 76

Question 76.
Suppose
\(f(x)=\left\{\begin{array}{cl}{a+b x,} & {x<1} \\{4,} & {x=1 \text { and }} \\{b-a x,} &{x>1}\end{array}\right.\)
\(\lim _{x \rightarrow 1} f(x)=f(1)\).what are possible value of a and b?
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 77

Question 77.
Let a1, a2,…, an be fixed real numbers and define a function
f(x) = (x – a1)(x – a2)…(x – an).
\(\text { What is } \lim _{x \rightarrow a_{1}} f(x) ? \text { For some } a \neq a_{1}, a_{2}, \dots, a_{n}\) \(\text { 1) }\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 78

KSEEB Solutions

Question 78.
\(\begin{aligned}&\text { If } f(x)=\left\{\begin{aligned}|x|+1, & x<0 \\0, & x=0 \text { for what value(s) of } \\|x|-1, & x>0\end{aligned}\right.\\&^{t} a^{\prime} \text { does } \lim _{x \rightarrow a} f(x) \text { exists? }\end{aligned}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 79

Question 79.
If the function f(x) satisfies \(\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi, \text { evaluate } \lim _{x \rightarrow 1} f(x)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 80

Question 80.
If \(f(x)=\left\{\begin{array}{cc}{m x^{2}+n,} & {x<0} \\{n x+m,} & {0 \leq x \leq 1} \\{n x^{3}+m,} & {x>1}\end{array}\right.\)for what integers m and n does both
\(\lim _{x \rightarrow 0} f(x)\)\(\lim _{x \rightarrow 1} f(x) \text { exist? }\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 81

Derivaties:

Question 81.
Define a derivative of f(x) at a point.
Ans :
Suppose f is a real valued function and ‘a’ is a point in its domain of definition. The derivative of f at ‘a’ is defined by
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 82

Question 82.
Find the derivative at x = 2 of the function f(x) = 3x.
Answer:
We have
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 83
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 84

Question 83.
Find the derivative of f(x) = x2 – 2 at x = 10.
Answer:
We have,
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 85

Question 84.
Find the derivative of 99x at x = 100.
Answer:
We have,
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 86

Question 85.
Find the derivative of x at x = 1.
Answer:
We have,
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 87

Question 86.
Find the derivative of the function f(x) = 2x2 + 3x – 5 at x = -1. Also prove that
f'(0) + 3f'(l) = 0.
Answer:
We have,
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 88
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 89

KSEEB Solutions

Question 87.
Find the derivative of sinx at x = 0.
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 90

Question 88.
Find the derivative of f(x) = 3 at x = 3
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 91

Question 89.
Define derivative of f(x) at x.
Answer:
Suppose f is a real valued function, the function defined by\(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\),wherever the limit exists is defined to be derivative of f at x and is denoted by f'(x). Thus \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)

Question 90.
Find the derivative of f(x) = 10x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 92

Question 91.
Find the derivative of the following : (by first principles)
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 93
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 94
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 95
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 96
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 97
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 98
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 99
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 100
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 101
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 102
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 103
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 104

KSEEB Solutions

Algebra of derivative of functions

Theorem

Let f and g be two functions such that their derivatives are defined in a common domain. Then

(i) Derivative of sum of two functions is sum of the derivatives of the functions.
\(\cfrac{d}{d x}[f(x)+g(x)]=\cfrac{d}{d x} f(x)+\cfrac{d}{d x} g(x)\)

(ii) Derivative of difference of two functions is difference of the derivatives of the functions
\(\cfrac{d}{d x}[f(x)-g(x)]=\cfrac{d}{d x} f(x)-\cfrac{d}{d x} g(x)\)

(iii) Derivative of product of two functions is given by the following product rule:
\(\begin{aligned}\cfrac{d}{d x}[f(x) \cdot g(x)] & \\&=f(x)\left[\cfrac{d}{d x}g(x)\right]+g(x)\left[\cfrac{d}{d x} f(x)\right]\end{aligned}\)

(iv) Derivative of the quotient of two functions is given by the following quotient rule:
\(\cfrac{d}{d x} \cfrac{f(x)}{g(x)}=\cfrac{g(x) \cfrac{d}{d x} f(x)-f(x) \cfrac{d}{d x} g(x)}{[g(x)]^{2}}\)
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 105

1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 106

I. Differentiate the following with respect to x

Question 1.
y = 25
Answer:
\(\cfrac{d y}{d x}=0\)

Question 2.
\(y=\cfrac{\pi}{4}\)
Answer:
\(\cfrac{d y}{d x}=0\)

Question 3.
y = 5 cos α, α is a constant.
Answer:
\(\cfrac{d y}{d x}=0\)

KSEEB Solutions

Question 4.
\(y=x^{6}\)
Answer:
\(\cfrac{d y}{d x}=6 x^{5}\)

Question 5.
\(y=x^{-5}\)
Answer:
\(\cfrac{d y}{d x}=-5 x^{-6}\)

Question 6.
\(y=5 x^{\frac{7}{2}}\)
Answer:
\(\cfrac{d y}{d x}=5 \cdot \cfrac{7}{2} x^{\cfrac{7}{2}-1}=\cfrac{35}{2} x^{\cfrac{5}{2}}\)

Question 7.
\(y=8 \cdot x^{\cfrac{5}{2}} x^{-\cfrac{5}{3}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 107

Question 8.
y = ( 2 + x)2
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 108

Question 9.
\(y=\frac{3}{x^{5}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 109

KSEEB Solutions

Question 10.
\(y=\left(x+\cfrac{1}{x}\right)^{2}, x \neq 0\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 110

Question 11.
\(y=\left(\sqrt{x}+\cfrac{1}{\sqrt{x}}\right)^{2}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 111

Question 12.
y = (ax)m +bm
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 112

Question 13.
y = x3+ 4x2 +7x + 2
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 113

KSEEB Solutions

Question 14.
\(y=3+4 x-7 x^{2}-\sqrt{2} x^{3}+\pi x^{4}-\cfrac{2}{5} x^{5}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 114

Question 15.
\(y=2 x^{\cfrac{1}{2}}+6 x^{\cfrac{1}{3}}+2 x^{\cfrac{3}{2}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 115

Question 16.
\(y=x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 116

Question 17.
\(y=(x-2)^{2}(2 x-3)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 117

Question 18.
\(y=\cfrac{(x+5)\left(2 x^{2}-1\right)}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 118

Question 19.
\(y=\left(2 x^{2}+3\right)\left(x^{2}-x+2\right)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 119

Question 20.
\(y=\cfrac{3 x^{7}+x^{5}-2 x^{4}+x-3}{x^{4}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 120

Question 21.
\(y=\cfrac{(2 x-1)\left(5 x^{\cfrac{1}{2}}+7\right)}{x^{\cfrac{1}{2}}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 121

KSEEB Solutions

Question 22.
\(y=\left(x-\cfrac{1}{x}\right)\left(x^{2}-\cfrac{1}{x^{2}}\right)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 122

Question 23.
\(y=\cfrac{2 x^{2}-3 x+1}{\sqrt{x}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 123

Question 24.
\(y=x^{4}+7 x^{3}+8 x^{2}+3 x+2+\sqrt{x}+\frac{1}{\sqrt{x}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 124

Question 25.
y = a(x -2)(x – 3) + b
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 125

Question 26.
y = a(x – 2) (x – b)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 126

Question 27.
y = (ax2+b)2
Answer:
y = (ax2 + b)2
= a2x4 + 2abx2 + b2
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 127

Question 28.
y = x + a
Answer:
\(\cfrac{d y}{d x}=1+0=1\)

Question 29.
\(y=4 \sqrt{x}-2\)
Answer:
\( \cfrac{d y}{d x}=4\left(\cfrac{1}{2} x^{-\cfrac{1}{2}}\right)-0=2 x^{-\cfrac{1}{2}}\)

Question 30.
\(y=(p x+q)\left(\frac{r}{x}+s\right)\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 128

Question 31.
\(y=\cfrac{a}{x^{4}}-\cfrac{b}{x^{2}}+\cos x\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 129

Question 32.
\(y=\cfrac{a}{x^{4}}-\cfrac{b}{x^{2}}+\cos x\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 130

Question 33.
y = sin (x  + a)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 131

KSEEB Solutions

Question 34.
For the function
\(\begin{aligned}&f(x)=\cfrac{x^{100}}{100}+\cfrac{x^{99}}{99}+\ldots+\cfrac{x^{2}}{2}+x+1 . \text { Prove that }\\&f^{\prime}(1)=100 f^{\prime}(0)\end{aligned}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 132
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 133

II. Using product rule,differentiate the following with respect to x:

Question 1.
y =x sin x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 134

Question 2.
y = x cos x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 135

Question 3.
y = x3 sin x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 136

Question 4.
y = (x – 2)(x + 3)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 137

Question 5.
y = sinx cosx
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 138

KSEEB Solutions

Question 6.
y = sec x tan x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 139

Question 7.
y = cosec x cot x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 140

Question 8.
y = x5 cot x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 141

Question 9.
y = xn tan x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 142

Question 10.
y = (x3 + x2 + 1)sin x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 143

Question 11.
y = (x2 -5x + 6)sec x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 144

Question 12.
y = (x2 +1)cosx
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 145

Question 13.
x = sin 2x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 146

KSEEB Solutions

Question 14.
y = (5x3 4- 3x -1)(x -1)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 147

Question 15.
y = x-3(5 + 3x)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 148

Question 16.
y = x5(3-6x-9)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 149

Question 17.
y = x-4(3 – 4-5)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 150

Question 18.
y = (x2 – 5x + 6)(x3 + 2)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 151

Question 19.
y = x4(5 sin x -3 cos x)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 152

Question 20.
y = (x+ sec x)(x – tanx)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 153

Question 21.
y = (5 – 4 cosx)(1 – 2tanx)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 154

Question 22.
y = (1 + 2 tan x)(5 + 4 cos x)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 155

Question 23.
y = (x + cos x)(x – tanx)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 156

Question 24.
y = (x + sec x)(x -tan x)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 157

KSEEB Solutions

Question 25.
y= (ax2 + sinx)(p + q cos x)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 158

III. Using quotient rule, differentiable with respect to x

Question 1.
y = cot x
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 159
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 160

Question 2.
\(y=\cfrac{x^{n}-a^{n}}{x-a}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 161

Question 3.
\(y=\cfrac{1+x^{2}}{1-x^{2}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 162

Question 4.
\(y=\cfrac{7 x+4}{4 x-7}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 163

Question 5.
\(y=\cfrac{x}{x+5}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 164

Question 6.
\(y=\cfrac{x-a}{x-b}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 165

Question 7.
\(y=\cfrac{2 x+3}{x-2}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 167

Question 8.
\(y=\cfrac{a x+b}{c x+d}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 168

Question 9.
\(y=\cfrac{1+\cfrac{1}{x}}{1-\cfrac{1}{x}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 169

Question 10.
\(y=\cfrac{x^{2}+5 x-6}{4 x^{2}-x+3}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 170

Question 11.
\(y=\cfrac{x^{2}-1}{x^{2}+7 x+1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 171

KSEEB Solutions

Question 12.
\(y=\cfrac{(x-1)(x-2)}{(x-3)(x-4)}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 172

Question 13.
\(y=\cfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 173
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 174

Question 14.
\(y=\cfrac{a x^{2}+b x+c}{p x^{2}+q x+r}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 175

Question 15.
\(y=\cfrac{a x+b}{p x^{2}+q x+r}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 176

Question 16.
\(y=\cfrac{p x^{2}+q x+r}{a x+b}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 177

Question 17.
\(y=\cfrac{1}{p x^{2}+q x+r}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 178

Question 18.
\(y=\cfrac{1}{a x^{2}+b x+c}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 179

Question 19.
\(y=\cfrac{\sec x+1}{\sec x-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 180

Question 20.
\(y=\cfrac{1-\tan x}{1+\tan x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 181

Question 21.
\(y=\cfrac{\sec x-1}{\sec x+1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 182

Question 22.
\(y=\cfrac{a+b \sin x}{c+d \cos x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 183

KSEEB Solutions

Question 23.
\(y=\cfrac{4 x+5 \sin x}{3 x+7 \cos x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 184

Question 24.
\(y=\cfrac{a+\sin x}{1+a \sin x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 185

Question 25.
\(y=\cfrac{x^{5}-\cos x}{\sin x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 186

Question 26.
\(y=\cfrac{x+\cos x}{\tan x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 187

Question 27.
\(y=\cfrac{\cos x}{1+\sin x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 188

Question 28.
\(y=\cfrac{\sin x+\cos x}{\sin x-\cos x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 189
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 190

Question 29.
\(y=\frac{x^{2} \cos \frac{\pi}{4}}{\sin x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 191

Question 30.
\(y=\cfrac{x^{2} \cos \cfrac{\pi}{4}}{\sin x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 192

Limits involving exponential and Logarithmic functions:

Question 1.
Define a logarithmic function.
Answer:
The logarithmic function expressed as loge: R+ → R is given by loge x = y, if and only if ey = x

KSEEB Solutions

Question 2.
Prove that \(\lim _{x \rightarrow 0} \cfrac{e^{x}-1}{x}=1\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 193

Question 3.
Prove that
\(\lim _{x \rightarrow 0} \cfrac{\log _{e}(1+x)}{x}=1\)
Answer:
\(\text { Let } \frac{\log _{e}(1+x)}{x}=y . \text { Then }\)
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 194

Question 4.
Compute
\(\lim _{x \rightarrow 0} \cfrac{e^{3 x}-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 195
Question 5.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{e^{4 x}-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 196

Question 6.
Compute
\(\lim _{x \rightarrow 0} \cfrac{e^{2+x}-e^{2}}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 197

Question 7.
Compute
\(\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 198

Question 8.
Compute
\(\lim _{x \rightarrow 0} \cfrac{e^{x}-\sin x-1}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 199

Question 9.
Evalute
\(\lim _{x \rightarrow 5} \cfrac{e^{x}-e^{5}}{x-5}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 200

Question 10.
Compute:
\(\lim _{x \rightarrow 3} \frac{e^{x}-e^{3}}{x-3}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 201

Question 11.
\(\lim _{x \rightarrow 0} \cfrac{x\left(e^{x}-1\right)}{1-\cos x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 202

Question 12.
Evaluate:
\(\lim _{x \rightarrow 0} \cfrac{\log _{e}(1+2 x)}{x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 203

Question 13.
Evaluate:
\(\lim _{x \rightarrow 0} \frac{\log \left(1+x^{3}\right)}{\sin ^{3} x}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 204

Question 14.
Evaluate
\(\lim _{x \rightarrow 1} \cfrac{\log _{e} x}{x-1}\)
Answer:
1st PUC Maths Question Bank Chapter 13 Limits and Derivatives 205

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