Are these practice questions for Limits and Derivatives based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 11 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Limits and Derivatives free?

Yes. Every chapter includes free practice questions for Class 11 - Science students to get you started.

What types of questions are included?

The practice set for Limits and Derivatives mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Limits and Derivatives for full coverage.

How many practice questions are there for Limits and Derivatives?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Limits and Derivatives Class 11 Mathematics Practice Questions with Answers

easy

Calculate the limit:

\lim_{x \to 4} \frac{x^2 - x - 12}{x - 4}

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easy

Calculate the limit:

\lim_{x \to 2} (3x^2 - x + 5)

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easy

State the formula for the derivative of a function

f(x)

at a point

x=a

using the first principle.

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easy

Calculate the derivative of the function

f(x) = 7x^3 - 4x^2 + 2x - 9

.

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easy

State the product rule for derivatives. If

u=f(x)

and

v=g(x)

, what is the formula for the derivative of their product

uv

?

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easy

Critique the following evaluation of a limit and justify why it is incorrect:

\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \frac{3^2 - 9}{3 - 3} = \frac{0}{0} = 1

.

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easy

Identify the standard derivatives for the following functions: (i)

f(x) = x^n

(for any positive integer

n

) (ii)

f(x) = \sin x

(iii)

f(x) = \cos x

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easy

Recall the standard limit formula for

\lim_{x \rightarrow a} \frac{x^n - a^n}{x - a}

.

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easy

Explain the condition required for the limit of a function

f(x)

to exist at a point

x=a

.

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easy

Define the Right Hand Limit of a function

f(x)

at a point

x=a

.

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easy

What is the derivative of a constant function

f(x) = c

, where

c

is a real number?

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medium

Calculate the derivative of

f(x) = x^3 \cos x

using the product rule.

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medium

Evaluate the derivative of

y = \frac{x \sin x}{1 + x}

and justify each major step by naming the differentiation rule used.

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medium

Solve for the limit:

\lim_{x \to 2} \frac{x^5 - 32}{x^2 - 4}

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medium

Formulate a rational function

f(x) = \frac{P(x)}{Q(x)}

, where

P(x)

and

Q(x)

are polynomials, such that

\lim_{x \to 1} f(x)

initially presents as the indeterminate form

\frac{0}{0}

but ultimately evaluates to 6.

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medium

Calculate the derivative of

f(x) = \frac{1}{2x+3}

from first principles.

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medium

Justify whether the function

f(x) = \begin{cases} x^2+1, & x \le 1 \ 3x-1, & x > 1 \end{cases}

is differentiable at

x=1

.

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medium

Calculate the derivative of

f(x) = \sec x - 10 \tan x

.

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medium

Evaluate the limit

\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{\cos(2x)}

and justify your method.

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medium

State the quotient rule for derivatives. If

u=f(x)

and

v=g(x)

, what is the formula for the derivative of their quotient

u/v

?

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medium

Calculate the derivative of

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

using the quotient rule.

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medium

Describe the relationship between the derivative of a function at a point and the tangent to the curve at that point.

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medium

Evaluate the limit:

\lim_{x \to 2} \left( \frac{1}{x-2} - \frac{4}{x^2-4} \right)

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medium

State the Sandwich Theorem for limits.

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medium

Summarize the intuitive idea of a derivative as explained through the example of a body's velocity. Explain what average velocity and instantaneous velocity represent in this context.

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medium

Formulate a single, non-piecewise rational function

f(x)

that is defined for all real numbers except

x=5

, and whose limit as

x \to 5

is 10.

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medium

Justify, by evaluating the left-hand and right-hand derivatives, why the function

f(x) = |x+1|

is not differentiable at

x=-1

.

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medium

Evaluate the limit:

\lim_{x \to 0} \frac{\sin(7x)}{3x}

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medium

List and explain the four main rules from the algebra of limits (Theorem 1). For each rule, let

\lim_{x \rightarrow a} f(x)

and

\lim_{x \rightarrow a} g(x)

exist.

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medium

List the two important trigonometric limits involving

\sin x

and

\cos x

as

x \rightarrow 0

.

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medium

Given that

\lim_{x \to 1} \frac{ax^2 + bx - 4}{x - 1} = 8

, formulate a system of equations to find the constants

a

and

b

. Justify your reasoning and solve for the constants.

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hard

Solve for the limit:

\lim_{x \to \pi} \frac{x - \pi}{\sin x}

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hard

Derive the formula for the derivative of

f(x) = \sin(x^2)

from first principles. Justify all key steps in your derivation, including the use of trigonometric identities and standard limits.

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hard

Derive the formula for the derivative of

f(x) = \sec x

from first principles.

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hard

Calculate the derivative of the function

f(x) = \frac{x \sin x}{1 + \cos x}

.

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hard

Analyze the function

f(x) = \begin{cases} x^2 - 1, & x < 1 \ a, & x = 1 \ b - x^3, & x > 1 \end{cases}

If

\lim_{x \to 1} f(x) = f(1)

, calculate the values of the constants

a

and

b

.

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hard

Propose a non-constant function

f(x)

such that the derivative of the product

x^2 f(x)

is

6x^2

.

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hard

Explain the concept of a rational function and describe how to find the limit of a rational function

f(x) = \frac{g(x)}{h(x)}

as

x \rightarrow a

, when

h(a) \neq 0

.

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hard

Calculate the derivative of

f(x) = x^2 - 4x

from first principles.

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hard

Design a polynomial function

f(x)

of the lowest possible degree that satisfies the following two conditions:

\lim_{x \to 0} \frac{f(x)}{x^2} = 5

and

\lim_{x \to 0} \frac{f(x)}{x^3}

does not exist.

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hard

List and describe the four main rules from the algebra of derivatives (Theorem 5). For each rule, let

u=f(x)

and

v=g(x)

.

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hard

Formulate a proof from first principles for the Product Rule of differentiation. That is, if

k(x) = f(x)g(x)

, prove that

k'(x) = f'(x)g(x) + f(x)g'(x)

, assuming

f

and

g

are differentiable.

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hard

Evaluate the limit:

\lim_{x \to 0} \frac{x \tan x}{1 - \cos(2x)}

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hard

Justify the steps required to evaluate the limit

\lim_{x \to 0} \frac{1 - \cos(4x)}{x^2}

.

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hard

Critique the following statement and justify your conclusion with a counterexample: 'If

\lim_{x \to a} f(x)

and

\lim_{x \to a} g(x)

both do not exist, then

\lim_{x \to a} [f(x) \cdot g(x)]

also cannot exist.'

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Limits and DerivativesClass 11 Mathematics Practice Questions