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

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

Question 1.
Explain the meaning of x → a .
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 .

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.

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.
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

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.

$$\text { We say } \lim _{x \rightarrow a^{+}} f(x)$$ 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.

Question 4.
Discuss the limits of the function
$$f(x)=\left\{\begin{array}{ll}{-1,} & {\text { if } x<0} \\{1,} & {\text { if } x>0}\end{array} \text { at } x=0\right.$$

∴ LHL ≠ RHL
Limit does not exist

Question 5.
Discuss the limit of the function f(x) = x +10 at x = 5.

Question 6.
Discuss the limit of f(x) = x3 at x = 1.

Question 7.
Find $$\lim _{x \rightarrow 2} f(x), \text { where } f(x)=3x$$

Question 8.
Find $$\lim _{x \rightarrow 2} f(x)$$,where f(x) = 3 a constant function

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

Question 10.
Find
f(x)=\left\{\begin{aligned}x-2, & x<0 \\0, & x=0 \\x+2, & x>0\end{aligned}\right.

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

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

Question 13.
Find $$\lim _{x \rightarrow-1}\left(1+x+x^{2}+\ldots+x^{10}\right)$$

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

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

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

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

Question 18.
Find
$$\lim _{x \rightarrow-1} \cfrac{x^{10}+x^{5}+1}{x-1}$$

Question 19.
Find
$$\lim _{x \rightarrow 0} \cfrac{(x+1)^{2}-1}{x}$$

Question 20.
Find
$$\lim _{x \rightarrow 2} \cfrac{3 x^{2}-x-10}{x^{2}-4}$$

Question 21.
Find
$$\lim _{x \rightarrow 3} \cfrac{x^{4}-81}{2 x^{2}-5 x-3}$$

Question 22.
Find
$$\lim _{x \rightarrow 0} \cfrac{a x+b}{c x+1}$$

Question 23.
Find
$$\lim _{x \rightarrow 1} \cfrac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c \neq 0$$

Question 24.
Find
$$\lim _{x \rightarrow 2} \cfrac{\cfrac{1}{x}+\cfrac{1}{2}}{x+2}$$

Question 25.
Find
$$\lim _{x \rightarrow 1} \cfrac{x^{2}+1}{x+100}$$

Question 26.
Find
$$\lim _{x \rightarrow 2} \cfrac{x^{3}-4 x^{2}+4 x}{x^{2}-4}$$

Question 27.
Find
$$\lim _{x \rightarrow 2} \cfrac{x^{2}-4}{x^{3}-4 x^{2}+4 x}$$

Question 28.
Find
$$\lim _{x \rightarrow 2} \cfrac{x^{3}-2 x^{2}}{x^{2}-5 x+6}$$

Question 29.
Find
$$\lim _{x \rightarrow 1}\left[\cfrac{x-2}{x^{2}-x}-\cfrac{1}{x^{3}-3 x^{2}+2 x}\right]$$

Question 30.
Prove that,for any positive integer n,
$$\lim _{x \rightarrow a} \cfrac{x^{n}-a^{n}}{x-a}=n\left(a^{n-1}\right)$$

Question 31.
Find
$$\lim _{x \rightarrow a} \cfrac{x^{3}-a^{3}}{x-a}$$
$$\lim _{x \rightarrow a} \cfrac{x^{3}-a^{3}}{x-a}=3 a^{2}$$

Question 32.
Find
$$\lim _{x \rightarrow 2} \cfrac{x^{7}-128}{x-2}$$

Question 33.
Find
$$\lim _{x \rightarrow 1} \cfrac{x^{3}-1}{x-1}$$

Question 34.
Find
$$\lim _{x \rightarrow a} \cfrac{x^{2 / 7}-a^{2 / 7}}{x-a}$$

Question 35.
Find
$$\lim _{x \rightarrow-a} \cfrac{x^{5}+a^{5}}{x+a}$$

Question 36.
Find
$$\lim _{x \rightarrow-1} \cfrac{x^{3}+1}{x+1}$$

Question 37.
Find
$$\lim _{x \rightarrow a} \cfrac{x \sqrt{x}-a \sqrt{a}}{x-a}$$

Question 38.
Find
$$\lim _{2 x \rightarrow-1} \cfrac{8 x^{3}+1}{2 x+1}$$

Question 39.
Find
$$\lim _{x \rightarrow 1} \cfrac{x^{15}-1}{x^{10}-1}$$

Question 40.
Find
$$\lim _{x \rightarrow 2} \cfrac{x^{3}-8}{x^{2}-4}$$

Question 41.
Find
$$\lim _{x \rightarrow 4} \cfrac{x^{3}-64}{x^{2}-16}$$

Question 42.
Find
$$\lim _{x \rightarrow 2} \cfrac{x^{10}-1024}{x^{5}-32}$$

Question 43.
Find
$$\lim _{x \rightarrow 9} \cfrac{x^{\cfrac{3}{2}}-27}{x-9}$$

Question 44.
Find
$$\lim _{x \rightarrow a} \cfrac{x^{\cfrac{2}{3}}-a^{\cfrac{2}{3}}}{x^{\cfrac{3}{4}}-a^{\cfrac{3}{4}}}$$

Question 45.
$$\lim _{x \rightarrow 0} \cfrac{\sqrt{1+x}-1}{x}$$

Question 46.
Find
$$\lim _{z \rightarrow 1} \cfrac{z^{\cfrac{1}{3}}-1}{z^{\cfrac{1}{6}}-1}$$

Question 47.
Find
$$\lim _{x \rightarrow 0} \cfrac{(1-x)^{n}-1}{x}$$

Question 48.
Find
$$\lim _{x \rightarrow 1} \cfrac{\left(x+x^{2}+x^{3}+\ldots+x^{n}\right)-n}{x-1}$$

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. $$\lim _{x \rightarrow a} f(x) \leq \lim _{x \rightarrow a} g(x)$$

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
$$\lim _{x \rightarrow 0} \cfrac{\sin x}{x}=1$$

Question 50.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\sin a x}{b x}$$

Question 51.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\sin a x}{\sin b x}, a, b \neq 0$$

Question 52.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\sin 4 x}{\sin 6 x}$$

Question 53.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\tan x}{x}$$

Question 54.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\tan 3 x}{\sin 2 x}$$

Question 55.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\tan 8 x}{\sin 2 x}$$

Question 56.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\sin 5 x}{\tan 3 x}$$

Question 57.
Evaluate:
$$\lim _{x \rightarrow \pi} \cfrac{\sin (\pi-x)}{\pi(\pi-x)}$$

Question 58.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\cos x}{\pi-x}$$

Question 59.
Evaluate:
$$\lim _{\theta \rightarrow 0} \cfrac{1-\cos 4 \theta}{1-\cos 6 \theta}$$

Question 60.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{1-\cos 5 x}{1-\cos 6 x}$$

Question 61.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{1-\cos 3 x}{x^{2}}$$

Question 62.
Evaluate:
$$\lim _{\theta \rightarrow 0} \cfrac{1-\cos \theta}{2 \theta^{2}}$$

Question 63.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{1-\cos x}{x}$$

Question 64.
Evaluate:
\begin{aligned}&\lim x \cdot \sec x\\&x \rightarrow 0\end{aligned}

Question 65.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{a x+x \cos x}{b \sin x}$$

Question 66.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\sin a x+b x}{a x+\sin b x}, a, b, a+b \neq 0$$

Question 67.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{1-\cos x}{\sin ^{2} x}$$

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

Question 69.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\tan 2 x+\sin 2 x}{x}$$

Question 70.
Evaluate:
$$\lim _{x \rightarrow 0}[\csc x-\cot x]$$

Question 71.
Evaluate:
$$\lim _{x \rightarrow \cfrac{\pi}{2}} \cfrac{\tan 2 x}{x-\cfrac{\pi}{2}}$$

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}

Question 73.
Find
$$\lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\{\begin{array}{cc}{x^{2}-1,} & {x \leq 1} \\{-x^{2}-1,} & {x>1}\end{array}\right.$$

Question 74.
Evaluate
$$\lim _{x \rightarrow 0} f(x), \text { where } f(x)=\left\{\begin{array}{cc}{\frac{|x|}{x},} & {x \neq 0} \\{0,} & {x=0}\end{array}\right.$$

Question 75.
Find
$$\lim _{x \rightarrow 5} f(x), \text { where } f(x)=|x|-5$$

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?

Question 77.
Let a1, a2,…, an be fixed real numbers and define a function
f(x) = (x – a1)(x – a2)…(x – an).
$$\text { What is } \lim _{x \rightarrow a_{1}} f(x) ? \text { For some } a \neq a_{1}, a_{2}, \dots, a_{n}$$ $$\text { 1) }$$

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}

Question 79.
If the function f(x) satisfies $$\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi, \text { evaluate } \lim _{x \rightarrow 1} f(x)$$

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
$$\lim _{x \rightarrow 0} f(x)$$$$\lim _{x \rightarrow 1} f(x) \text { exist? }$$

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

Question 82.
Find the derivative at x = 2 of the function f(x) = 3x.
We have

Question 83.
Find the derivative of f(x) = x2 – 2 at x = 10.
We have,

Question 84.
Find the derivative of 99x at x = 100.
We have,

Question 85.
Find the derivative of x at x = 1.
We have,

Question 86.
Find the derivative of the function f(x) = 2x2 + 3x – 5 at x = -1. Also prove that
f'(0) + 3f'(l) = 0.
We have,

Question 87.
Find the derivative of sinx at x = 0.

Question 88.
Find the derivative of f(x) = 3 at x = 3

Question 89.
Define derivative of f(x) at x.
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

Question 91.
Find the derivative of the following : (by first principles)

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:
$$\cfrac{d}{d x} \cfrac{f(x)}{g(x)}=\cfrac{g(x) \cfrac{d}{d x} f(x)-f(x) \cfrac{d}{d x} g(x)}{[g(x)]^{2}}$$

I. Differentiate the following with respect to x

Question 1.
y = 25
$$\cfrac{d y}{d x}=0$$

Question 2.
$$y=\cfrac{\pi}{4}$$
$$\cfrac{d y}{d x}=0$$

Question 3.
y = 5 cos α, α is a constant.
$$\cfrac{d y}{d x}=0$$

Question 4.
$$y=x^{6}$$
$$\cfrac{d y}{d x}=6 x^{5}$$

Question 5.
$$y=x^{-5}$$
$$\cfrac{d y}{d x}=-5 x^{-6}$$

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

Question 7.
$$y=8 \cdot x^{\cfrac{5}{2}} x^{-\cfrac{5}{3}}$$

Question 8.
y = ( 2 + x)2

Question 9.
$$y=\frac{3}{x^{5}}$$

Question 10.
$$y=\left(x+\cfrac{1}{x}\right)^{2}, x \neq 0$$

Question 11.
$$y=\left(\sqrt{x}+\cfrac{1}{\sqrt{x}}\right)^{2}$$

Question 12.
y = (ax)m +bm

Question 13.
y = x3+ 4x2 +7x + 2

Question 14.
$$y=3+4 x-7 x^{2}-\sqrt{2} x^{3}+\pi x^{4}-\cfrac{2}{5} x^{5}$$

Question 15.
$$y=2 x^{\cfrac{1}{2}}+6 x^{\cfrac{1}{3}}+2 x^{\cfrac{3}{2}}$$

Question 16.
$$y=x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}$$

Question 17.
$$y=(x-2)^{2}(2 x-3)$$

Question 18.
$$y=\cfrac{(x+5)\left(2 x^{2}-1\right)}{x}$$

Question 19.
$$y=\left(2 x^{2}+3\right)\left(x^{2}-x+2\right)$$

Question 20.
$$y=\cfrac{3 x^{7}+x^{5}-2 x^{4}+x-3}{x^{4}}$$

Question 21.
$$y=\cfrac{(2 x-1)\left(5 x^{\cfrac{1}{2}}+7\right)}{x^{\cfrac{1}{2}}}$$

Question 22.
$$y=\left(x-\cfrac{1}{x}\right)\left(x^{2}-\cfrac{1}{x^{2}}\right)$$

Question 23.
$$y=\cfrac{2 x^{2}-3 x+1}{\sqrt{x}}$$

Question 24.
$$y=x^{4}+7 x^{3}+8 x^{2}+3 x+2+\sqrt{x}+\frac{1}{\sqrt{x}}$$

Question 25.
y = a(x -2)(x – 3) + b

Question 26.
y = a(x – 2) (x – b)

Question 27.
y = (ax2+b)2
y = (ax2 + b)2
= a2x4 + 2abx2 + b2

Question 28.
y = x + a
$$\cfrac{d y}{d x}=1+0=1$$

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

Question 30.
$$y=(p x+q)\left(\frac{r}{x}+s\right)$$

Question 31.
$$y=\cfrac{a}{x^{4}}-\cfrac{b}{x^{2}}+\cos x$$

Question 32.
$$y=\cfrac{a}{x^{4}}-\cfrac{b}{x^{2}}+\cos x$$

Question 33.
y = sin (x  + a)

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}

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

Question 1.
y =x sin x

Question 2.
y = x cos x

Question 3.
y = x3 sin x

Question 4.
y = (x – 2)(x + 3)

Question 5.
y = sinx cosx

Question 6.
y = sec x tan x

Question 7.
y = cosec x cot x

Question 8.
y = x5 cot x

Question 9.
y = xn tan x

Question 10.
y = (x3 + x2 + 1)sin x

Question 11.
y = (x2 -5x + 6)sec x

Question 12.
y = (x2 +1)cosx

Question 13.
x = sin 2x

Question 14.
y = (5x3 4- 3x -1)(x -1)

Question 15.
y = x-3(5 + 3x)

Question 16.
y = x5(3-6x-9)

Question 17.
y = x-4(3 – 4-5)

Question 18.
y = (x2 – 5x + 6)(x3 + 2)

Question 19.
y = x4(5 sin x -3 cos x)

Question 20.
y = (x+ sec x)(x – tanx)

Question 21.
y = (5 – 4 cosx)(1 – 2tanx)

Question 22.
y = (1 + 2 tan x)(5 + 4 cos x)

Question 23.
y = (x + cos x)(x – tanx)

Question 24.
y = (x + sec x)(x -tan x)

Question 25.
y= (ax2 + sinx)(p + q cos x)

III. Using quotient rule, differentiable with respect to x

Question 1.
y = cot x

Question 2.
$$y=\cfrac{x^{n}-a^{n}}{x-a}$$

Question 3.
$$y=\cfrac{1+x^{2}}{1-x^{2}}$$

Question 4.
$$y=\cfrac{7 x+4}{4 x-7}$$

Question 5.
$$y=\cfrac{x}{x+5}$$

Question 6.
$$y=\cfrac{x-a}{x-b}$$

Question 7.
$$y=\cfrac{2 x+3}{x-2}$$

Question 8.
$$y=\cfrac{a x+b}{c x+d}$$

Question 9.
$$y=\cfrac{1+\cfrac{1}{x}}{1-\cfrac{1}{x}}$$

Question 10.
$$y=\cfrac{x^{2}+5 x-6}{4 x^{2}-x+3}$$

Question 11.
$$y=\cfrac{x^{2}-1}{x^{2}+7 x+1}$$

Question 12.
$$y=\cfrac{(x-1)(x-2)}{(x-3)(x-4)}$$

Question 13.
$$y=\cfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}$$

Question 14.
$$y=\cfrac{a x^{2}+b x+c}{p x^{2}+q x+r}$$

Question 15.
$$y=\cfrac{a x+b}{p x^{2}+q x+r}$$

Question 16.
$$y=\cfrac{p x^{2}+q x+r}{a x+b}$$

Question 17.
$$y=\cfrac{1}{p x^{2}+q x+r}$$

Question 18.
$$y=\cfrac{1}{a x^{2}+b x+c}$$

Question 19.
$$y=\cfrac{\sec x+1}{\sec x-1}$$

Question 20.
$$y=\cfrac{1-\tan x}{1+\tan x}$$

Question 21.
$$y=\cfrac{\sec x-1}{\sec x+1}$$

Question 22.
$$y=\cfrac{a+b \sin x}{c+d \cos x}$$

Question 23.
$$y=\cfrac{4 x+5 \sin x}{3 x+7 \cos x}$$

Question 24.
$$y=\cfrac{a+\sin x}{1+a \sin x}$$

Question 25.
$$y=\cfrac{x^{5}-\cos x}{\sin x}$$

Question 26.
$$y=\cfrac{x+\cos x}{\tan x}$$

Question 27.
$$y=\cfrac{\cos x}{1+\sin x}$$

Question 28.
$$y=\cfrac{\sin x+\cos x}{\sin x-\cos x}$$

Question 29.
$$y=\frac{x^{2} \cos \frac{\pi}{4}}{\sin x}$$

Question 30.
$$y=\cfrac{x^{2} \cos \cfrac{\pi}{4}}{\sin x}$$

Limits involving exponential and Logarithmic functions:

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

Question 2.
Prove that $$\lim _{x \rightarrow 0} \cfrac{e^{x}-1}{x}=1$$

Question 3.
Prove that
$$\lim _{x \rightarrow 0} \cfrac{\log _{e}(1+x)}{x}=1$$
$$\text { Let } \frac{\log _{e}(1+x)}{x}=y . \text { Then }$$

Question 4.
Compute
$$\lim _{x \rightarrow 0} \cfrac{e^{3 x}-1}{x}$$

Question 5.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{e^{4 x}-1}{x}$$

Question 6.
Compute
$$\lim _{x \rightarrow 0} \cfrac{e^{2+x}-e^{2}}{x}$$

Question 7.
Compute
$$\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}$$

Question 8.
Compute
$$\lim _{x \rightarrow 0} \cfrac{e^{x}-\sin x-1}{x}$$

Question 9.
Evalute
$$\lim _{x \rightarrow 5} \cfrac{e^{x}-e^{5}}{x-5}$$

Question 10.
Compute:
$$\lim _{x \rightarrow 3} \frac{e^{x}-e^{3}}{x-3}$$

Question 11.
$$\lim _{x \rightarrow 0} \cfrac{x\left(e^{x}-1\right)}{1-\cos x}$$

Question 12.
Evaluate:
$$\lim _{x \rightarrow 0} \cfrac{\log _{e}(1+2 x)}{x}$$
$$\lim _{x \rightarrow 0} \frac{\log \left(1+x^{3}\right)}{\sin ^{3} x}$$
$$\lim _{x \rightarrow 1} \cfrac{\log _{e} x}{x-1}$$