Important Questions for Class 11 Maths Chapter 13: Limits and Derivatives

Important Questions for Class 11 Maths Chapter 13 Limits and Derivatives are provided in the article. Limits and derivatives are the initial concepts in mathematics prior to calculus. A limit is defined as a value that a function approaches as the input and produces some value. It is used to define integrals, derivatives and continuity. Limit of a function is represented as:

f(n)=L

A derivative refers to the instantaneous rate of change of a function with respect to a variable. The process of finding derivatives is known as differentiation. It helps to investigate the nature of an amount moment by moment. The derivative of a function is represented as:

\(\frac{f(x+h)-f(x)}{h} \)

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Very short answer questions [1 Mark Question]

Ques. Evaluate the function \(\frac{(x^2-9)}{x-3} \) . [CBSE 2016]

Ans. Here, the limit of the given function is in 00.

Now, by representing the numerator as the product of two terms, we get

 \(\frac{(x+3)(x-3)}{x-3} \)

= 3 + 3

= 6

Ques. Evaluate the function \(\frac{sinsin ax}{bx} \). [CBSE IE 2015,2013]

Ans. \(\frac{sinsin ax}{bx} \)=\(\frac{sinsin ax}{ax} \) X  \(\frac{a}{b} \)

= 1×\(\frac{a}{b} \) ? \(\frac{sinsinx}{x} \) =1

= ab

Ques. Find the derivative of 2^x with respect to x.

Ans. Assume the given expression as y=2^x

By differentiating on both sides with respect to x we get,

\(\frac{dy}{dx} \) = ddx(2^x)

\(\frac{dy}{dx} \)= 2^xlog log2

Ques. Find the derivative of \(\sqrt{sinsin2x} \) with respect to x.

Ans. By using the chain rule of differentiation the derivative of given function is expressed by,

=\(\frac{d}{dx} \)\(\sqrt{sinsin2x} \) = \(\frac{1}{2\sqrt{sinsin2x}} \) \(\frac{d}{dx} \)sin sin2x

= \(\frac{1}{2\sqrt{sinsin2x}} \)× 2cos cos2x

=  \(\frac{coscos 2x}{\sqrt{coscos 2x}} \)

Ques. Evaluates the function \(\frac{x^n-a^n}{x-a} \)xn-anx-a [CBSE 2013]

Ans. By differentiating the numerator and denominator with respect to x, we get

\(\frac{x^n-a^n}{x-a} \) = \(\frac{nx^{n-1}}{1} \)

=na^n-1

Ques. What is the derivative of \(\frac{2^x}{x} \)

Ans. By using the u/v formula of differentiating we get,

Ques. Find the value of \(\frac{dy}{dx} \) if the expression is y=esin x .

Ans. By differentiating the given expression both sides we get,

\(\frac{dy}{dx} \)= \(\frac{d}{dx} \)(e^{sinsin x} )

= e^sinsin  x cos x

=cos x e^{sin x} .

Ques. Differentiate xsinsin x with respect to x. [CBSE IE 2019]

Ans. By using the chain rule of differentiation we get,

\(\frac{d}{dx} \)(xsin x) = x(\(\frac{d}{dx} \)(sin sin x) )+ sin sinx \(\frac{d}{dx} \)(x)

=x(cos cos x) + sin sinx(1)

=x cos cosx + sin sinx

Ques. Evaluate the expression \(\frac{x^2+1}{x+100} \)

Ans. By substituting x=1 in the expression we get,

\(\frac{x^2+1}{x+100} \) = \(\frac{1+1}{1+100} \)

=2/101

Ques. Evaluate the expression \(\frac{tantan{\pi}x}{x+2} \)

Ans. By using differentiating the numerator and denominator with respect to x we get,

\(\frac{tantan{\pi}x}{x+2} \) = \(\frac{{\pi}sec^2({\pi}x)}{1} \)

=πsec²(-2π)

=π

Ques. Find the derivative of expression 1+x+x²+x³+…+x⁵0 for x=1.

Ans. By the power rule of differentiation we get,

f'(x)=1+2x+3x²+4x³+…+50x⁵⁰

By substituting x=1 in the above derivative we get,

f'(1)=1+2+3+…+50

=50(50+1)/2

=25×51

=1275

Ques. If the limit value is \(\frac{x^7+a^7}{x+a} \) , then find the value of variable a.

Ans. By substituting x=a in the expression we get,

\(\frac{x^7+a^7}{x+a} \)=7

\(\frac{a^7+a^7}{a+a} \)=7

a⁶=7

⇒a=⁶√7

Also read:

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Short answer questions [2 Marks Question]

Ques. Evaluate the expression [cosec x-cot x] .

Ans. By rewriting cosec x and cot x in terms of cos x and sin x we get,

=0

Ques. Find the derivative of expression x^-3(5+3x).

Ans. By multiplying x-3 to each term we get,

\(\frac{d}{dx} \)x^-3(5+3x)= \(\frac{d}{dx} \)[5x^-3+3x^-2]

Now, by differentiating the equation we get

\(\frac{d}{dx} \)x^-3(5+3x)=-15x^-4 - 6x^-3

\(\frac{d}{dx} \)[5x^-3+3x^-2]= -\(\frac{15}{x^2} \)-\(\frac{6}{x^3} \)

Ques. Find the derivative of f(x) = x³ using the first principle.

Ans. By definition of first principle,

f'x = \(\frac{f(x+h)-f(x)}{h} \)

By substituting f(x) = x³ in the above equation we get,

f'x = \(\frac{(x+h)^3-(x)^3}{h} \)

f'x = \(\frac{x^3+h^3+3xh(x+h)-x^3}{h} \)

f'x = h²+3x(x+h)

Substituting h=0 we get,

f'x=3x²

Ques. Find the derivative of \({e^{\sqrt{cotx}}} \).

Ans. By using the chain rule of differentiation, the derivative of the given expression is

Ques. Prove that the value of expression\(\frac{e^x-1}{x} \) is 1.

Ans.

By substituting the limit we get,

= 1 + 0

=1

Ques. Evaluate the function \(\frac{(1+cos 2x)}{(\pi-2x)^2} \) .

Ans. 

Also read:

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Long answer questions [3 Marks Question]

Ques. Evaluate the function \(\frac{(2x-3)(\sqrt{x}-1)}{2x^2+x-3} \) .

Ans. Represent the denominator as the product of two factors

\(\frac{(2x-3)(\sqrt{x}-1)}{2x^2+x-3} \) = \(\frac{(2x-3)(\sqrt{x}-1)}{(2x+3)(x-1)} \)

By using the formula for the expression x-1 in the denominator we get,

\(\frac{(2x-3)(\sqrt{x}-1)}{2x^2+x-3} \) = \(\frac{(2x-3)(\sqrt{x}-1)}{(2x+3)(\sqrt{x}-1)(\sqrt{x}+1)} \)

\(\frac{(2x-3)}{(2x+3)(\sqrt{x}+1)} \)

Now, substitute the value x=1 in the expression

=\(\frac{(2-3)}{(2+3)(1+1)} \)

=-1/10

Ques. If the expression is y= \(\frac{1-tantanx}{1+tantanx} \), then show that the derivative \(\frac{dy}{dx} \)= \(\frac{-2}{1+sinsin2x} \)

Ans. By differentiating the given expression on both sides we get,

Hence proved.

Ques. Let f(x) = {a+bx , x<1 4 , x=1 b-ax ,x>1 and f(x) = f(1) , then find the possible value of a and b. [IMP-2011]

Ans. We know that f(x) = lim_x→a^-f(x)

By using the above equation for the given function at x=1 we get,

lim_x→1⁺ f(x) = f(1) = 4…………….(1)

lim_x→1^- f(x) = f(1) = 4…………….(2)

For x>1 we have the function as f(x) = a+bx

Now, by using equation (1) we get,

lim_x→1⁺ f(x) = lim_x→1⁺(a+bx)

= a+b=4……………………(3)

Now, for x<1 we have the function as f(x) = b - ax

Now, by using equation (2) we get,

lim_x→a^-f(x) = lim_x→a^-(b-ax)

=a-b=4……………………(4)

Now, by adding equation (3) and (4) we get,

4+4 = (a+b) + (b-a)

8=2b

b=4

Similarly, the value of a=0

Ques. Find the derivative of f(x) = secsec x by the first principle. [IMP-2010]

Ans. The given function is f(x) = secsec x

We can write it as 

f(x+h) =secsec(x+h)

By definition of first principle, we have,

Ques. Find the derivative of sin²x using product rule. [CBSE IE 2016]

Ans. Given function is y=sin²x

By writing it as product of two function we get,

⇒y=sin x × sin x

Differentiate on both sides using product rule.

\(\frac{dy}{dx} \) = \(\frac{d}{dx} \)(sin x × sin x )

=sinsinx\(\frac{d}{dx} \)sinsinx + sinx\(\frac{d}{dx} \)sinx

=sinxcoscos x + sin sin xcos x

=sin2x

Ques. Find the derivative of \(\frac{x^n-a^n}{x-a} \)

Ans. By differentiating using the quotient rule

Also read:

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Very long answer questions [5 Marks Question]

Ques. Differentiate x using first principle. [IMP-2012]

Ans. The given function is f(x) = x

We can write it as 

f(x+h) = cosec (x+h)

Ques. Evaluate \(\frac{(a+h)^2sin(a+h)-a^2sina}{h} \)

Ans.

Ques. Find derivative of function [CBSE IE 2020]

1. \(\frac{xsinsinx}{1+coscosx} \)
2. (ax+b)(cx+d)²

Ans.

Ques. Differentiate the given functions [CBSE 2016,2018]

1. \(\frac{a}{x^4}\) - \(\frac{b}{x^2}\) + cos x
2. (x + cos x)(x – tan x)

Ans.

Also check: