Bank Guides Notes Mocks Board Questions / CBSE Class 11 / Mathematics Limits and Derivatives 126 CBSE Class 11 Mathematics practice questions with answers and worked solutions.
Difficulty: 42 easy · 75 moderate · 9 hard. Most-asked subtopics: Algebra of Derivatives and Standard Formulas (58), Derivatives from First Principles (24), Limits of Polynomials and Rational Functions (23). Updated 18 August 2026. Find the limits:
NCERT textbook · 12.1 Eg.1(i) Easy Add #1 · Limits of Polynomials and Rational Functions
lim x → 1 [ x 3 − x 2 + 1 ] \lim_{x \to 1}\left[x^3 - x^2 + 1\right] lim x → 1 [ x 3 − x 2 + 1 ] NCERT textbook · 12.1 Eg.1(ii) Easy Add #2 · Limits of Polynomials and Rational Functions
lim x → 3 [ x ( x + 1 ) ] \lim_{x \to 3}\left[x(x+1)\right] lim x → 3 [ x ( x + 1 ) ] NCERT textbook · 12.1 Eg.1(iii) Easy Add #3 · Limits of Polynomials and Rational Functions
lim x → − 1 [ 1 + x + x 2 + … + x 10 ] \lim_{x \to -1}\left[1 + x + x^2 + \ldots + x^{10}\right] lim x → − 1 [ 1 + x + x 2 + … + x 10 ] Find the limits:
NCERT textbook · 12.1 Eg.2(i) Easy Add #4 · Limits of Polynomials and Rational Functions
lim x → 1 [ x 2 + 1 x + 100 ] \lim_{x \to 1}\left[\frac{x^2 + 1}{x + 100}\right] lim x → 1 [ x + 100 x 2 + 1 ] NCERT textbook · 12.1 Eg.2(ii) Moderate Add #5 · Limits of Polynomials and Rational Functions
lim x → 2 [ x 3 − 4 x 2 + 4 x x 2 − 4 ] \lim_{x \to 2}\left[\frac{x^3 - 4x^2 + 4x}{x^2 - 4}\right] lim x → 2 [ x 2 − 4 x 3 − 4 x 2 + 4 x ] NCERT textbook · 12.1 Eg.2(iii) Moderate Add #6 · Limits of Polynomials and Rational Functions
lim x → 2 [ x 2 − 4 x 3 − 4 x 2 + 4 x ] \lim_{x \to 2}\left[\frac{x^2 - 4}{x^3 - 4x^2 + 4x}\right] lim x → 2 [ x 3 − 4 x 2 + 4 x x 2 − 4 ] NCERT textbook · 12.1 Eg.2(iv) Moderate Add #7 · Limits of Polynomials and Rational Functions
lim x → 2 [ x 3 − 2 x 2 x 2 − 5 x + 6 ] \lim_{x \to 2}\left[\frac{x^3 - 2x^2}{x^2 - 5x + 6}\right] lim x → 2 [ x 2 − 5 x + 6 x 3 − 2 x 2 ] NCERT textbook · 12.1 Eg.2(v) Hard Add #8 · Limits of Polynomials and Rational Functions
lim x → 1 [ x − 2 x 2 − x − 1 x 3 − 3 x 2 + 2 x ] \lim_{x \to 1}\left[\frac{x-2}{x^2 - x} - \frac{1}{x^3 - 3x^2 + 2x}\right] lim x → 1 [ x 2 − x x − 2 − x 3 − 3 x 2 + 2 x 1 ] Done a few? Here’s your next move
Evaluate:
NCERT textbook · 12.1 Eg.3(i) Moderate Add #9 · Limits of Polynomials and Rational Functions
lim x → 1 x 15 − 1 x 10 − 1 \lim_{x \to 1}\frac{x^{15} - 1}{x^{10} - 1} lim x → 1 x 10 − 1 x 15 − 1 NCERT textbook · 12.1 Eg.3(ii) Moderate Add #10 · Limits of Polynomials and Rational Functions
lim x → 0 1 + x − 1 x \lim_{x \to 0}\frac{\sqrt{1+x} - 1}{x} lim x → 0 x 1 + x − 1 Evaluate:
NCERT textbook · 12.1 Eg.4(i) Moderate Add #11 · Limits of Trigonometric Functions
lim x → 0 sin 4 x sin 2 x \lim_{x \to 0}\frac{\sin 4x}{\sin 2x} lim x → 0 s i n 2 x s i n 4 x NCERT textbook · 12.1 Eg.4(ii) Easy Add #12 · Limits of Trigonometric Functions
lim x → 0 tan x x \lim_{x \to 0}\frac{\tan x}{x} lim x → 0 x t a n x NCERT textbook · Ex 12.1 Q1 Easy Add #13 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 3 x + 3 \lim_{x \to 3} x + 3 lim x → 3 x + 3 . NCERT textbook · Ex 12.1 Q2 Easy Add #14 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → π ( x − 22 7 ) \lim_{x \to \pi} \left( x - \frac{22}{7} \right) lim x → π ( x − 7 22 ) . NCERT textbook · Ex 12.1 Q3 Easy Add #15 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim r → 1 π r 2 \lim_{r \to 1} \pi r^2 lim r → 1 π r 2 . NCERT textbook · Ex 12.1 Q4 Easy Add #16 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 4 4 x + 3 x − 2 \lim_{x \to 4} \frac{4x + 3}{x - 2} lim x → 4 x − 2 4 x + 3 . NCERT textbook · Ex 12.1 Q5 Easy Add #17 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → − 1 x 10 + x 5 + 1 x − 1 \lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1} lim x → − 1 x − 1 x 10 + x 5 + 1 . NCERT textbook · Ex 12.1 Q6 Moderate Add #18 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 0 ( x + 1 ) 5 − 1 x \lim_{x \to 0} \frac{(x + 1)^5 - 1}{x} lim x → 0 x ( x + 1 ) 5 − 1 . NCERT textbook · Ex 12.1 Q7 Moderate Add #19 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 2 3 x 2 − x − 10 x 2 − 4 \lim_{x \to 2} \frac{3x^2 - x - 10}{x^2 - 4} lim x → 2 x 2 − 4 3 x 2 − x − 10 . NCERT textbook · Ex 12.1 Q8 Moderate Add #20 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 3 x 4 − 81 2 x 2 − 5 x − 3 \lim_{x \to 3} \frac{x^4 - 81}{2x^2 - 5x - 3} lim x → 3 2 x 2 − 5 x − 3 x 4 − 81 . NCERT textbook · Ex 12.1 Q9 Easy Add #21 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 0 a x + b c x + 1 \lim_{x \to 0} \frac{ax + b}{cx + 1} lim x → 0 c x + 1 a x + b . NCERT textbook · Ex 12.1 Q10 Moderate Add #22 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim z → 1 z 1 3 − 1 z 1 6 − 1 \lim_{z \to 1} \frac{z^{\frac{1}{3}} - 1}{z^{\frac{1}{6}} - 1} lim z → 1 z 6 1 − 1 z 3 1 − 1 . NCERT textbook · Ex 12.1 Q11 Easy Add #23 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → 1 a x 2 + b x + c c x 2 + b x + a \lim_{x \to 1} \frac{ax^2 + bx + c}{cx^2 + bx + a} lim x → 1 c x 2 + b x + a a x 2 + b x + c , a + b + c ≠ 0 a + b + c \neq 0 a + b + c = 0 . NCERT textbook · Ex 12.1 Q12 Moderate Add #24 · Limits of Polynomials and Rational Functions
Evaluate the following limit: lim x → − 2 1 x + 1 2 x + 2 \lim_{x \to -2} \dfrac{\frac{1}{x} + \frac{1}{2}}{x + 2} lim x → − 2 x + 2 x 1 + 2 1 . NCERT textbook · Ex 12.1 Q13 Easy Add #25 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 sin a x b x \lim_{x \to 0} \frac{\sin ax}{bx} lim x → 0 b x s i n a x . NCERT textbook · Ex 12.1 Q14 Moderate Add #26 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 sin a x sin b x \lim_{x \to 0} \frac{\sin ax}{\sin bx} lim x → 0 s i n b x s i n a x , a , b ≠ 0 a, b \neq 0 a , b = 0 . NCERT textbook · Ex 12.1 Q15 Moderate Add #27 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → π sin ( π − x ) π ( π − x ) \lim_{x \to \pi} \frac{\sin(\pi - x)}{\pi(\pi - x)} lim x → π π ( π − x ) s i n ( π − x ) . NCERT textbook · Ex 12.1 Q16 Easy Add #28 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 cos x π − x \lim_{x \to 0} \frac{\cos x}{\pi - x} lim x → 0 π − x c o s x . NCERT textbook · Ex 12.1 Q17 Moderate Add #29 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 cos 2 x − 1 cos x − 1 \lim_{x \to 0} \frac{\cos 2x - 1}{\cos x - 1} lim x → 0 c o s x − 1 c o s 2 x − 1 . NCERT textbook · Ex 12.1 Q18 Moderate Add #30 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 a x + x cos x b sin x \lim_{x \to 0} \frac{ax + x \cos x}{b \sin x} lim x → 0 b s i n x a x + x c o s x . NCERT textbook · Ex 12.1 Q19 Easy Add #31 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 x sec x \lim_{x \to 0} x \sec x lim x → 0 x sec x . NCERT textbook · Ex 12.1 Q20 Moderate Add #32 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 sin a x + b x a x + sin b x \lim_{x \to 0} \frac{\sin ax + bx}{ax + \sin bx} lim x → 0 a x + s i n b x s i n a x + b x , a , b , a + b ≠ 0 a, b, a + b \neq 0 a , b , a + b = 0 . NCERT textbook · Ex 12.1 Q21 Moderate Add #33 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → 0 ( cosec x − cot x ) \lim_{x \to 0} (\operatorname{cosec} x - \cot x) lim x → 0 ( cosec x − cot x ) . NCERT textbook · Ex 12.1 Q22 Hard Add #34 · Limits of Trigonometric Functions
Evaluate the following limit: lim x → π 2 tan 2 x x − π 2 \lim_{x \to \frac{\pi}{2}} \dfrac{\tan 2x}{x - \dfrac{\pi}{2}} lim x → 2 π x − 2 π tan 2 x . NCERT textbook · Ex 12.1 Q23 Easy Add #35 · Limits and the Algebra of Limits
Find lim x → 0 f ( x ) \lim_{x \to 0} f(x) lim x → 0 f ( x ) and lim x → 1 f ( x ) \lim_{x \to 1} f(x) lim x → 1 f ( x ) , where f ( x ) = { 2 x + 3 , x ≤ 0 3 ( x + 1 ) , x > 0 f(x) = \begin{cases} 2x + 3, & x \le 0 \\ 3(x + 1), & x > 0 \end{cases} f ( x ) = { 2 x + 3 , 3 ( x + 1 ) , x ≤ 0 x > 0 NCERT textbook · Ex 12.1 Q24 Moderate Add #36 · Limits and the Algebra of Limits
Find lim x → 1 f ( x ) \lim_{x \to 1} f(x) lim x → 1 f ( x ) , where f ( x ) = { x 2 − 1 , x ≤ 1 − x 2 − 1 , x > 1 f(x) = \begin{cases} x^2 - 1, & x \le 1 \\ -x^2 - 1, & x > 1 \end{cases} f ( x ) = { x 2 − 1 , − x 2 − 1 , x ≤ 1 x > 1 NCERT textbook · Ex 12.1 Q25 Moderate Add #37 · Limits and the Algebra of Limits
Evaluate lim x → 0 f ( x ) \lim_{x \to 0} f(x) lim x → 0 f ( x ) , where f ( x ) = { ∣ x ∣ x , x ≠ 0 0 , x = 0 f(x) = \begin{cases} \dfrac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases} f ( x ) = ⎩ ⎨ ⎧ x ∣ x ∣ , 0 , x = 0 x = 0 NCERT textbook · Ex 12.1 Q26 Moderate Add #38 · Limits and the Algebra of Limits
Find lim x → 0 f ( x ) \lim_{x \to 0} f(x) lim x → 0 f ( x ) , where f ( x ) = { x ∣ x ∣ , x ≠ 0 0 , x = 0 f(x) = \begin{cases} \dfrac{x}{|x|}, & x \neq 0 \\ 0, & x = 0 \end{cases} f ( x ) = ⎩ ⎨ ⎧ ∣ x ∣ x , 0 , x = 0 x = 0 NCERT textbook · Ex 12.1 Q27 Easy Add #39 · Limits and the Algebra of Limits
Find lim x → 5 f ( x ) \lim_{x \to 5} f(x) lim x → 5 f ( x ) , where f ( x ) = ∣ x ∣ − 5 f(x) = |x| - 5 f ( x ) = ∣ x ∣ − 5 . NCERT textbook · Ex 12.1 Q28 Moderate Add #40 · Limits and the Algebra of Limits
Suppose f ( x ) = { a + b x , x < 1 4 , x = 1 b − a x , x > 1 f(x) = \begin{cases} a + bx, & x < 1 \\ 4, & x = 1 \\ b - ax, & x > 1 \end{cases} f ( x ) = ⎩ ⎨ ⎧ a + b x , 4 , b − a x , x < 1 x = 1 x > 1 and if lim x → 1 f ( x ) = f ( 1 ) \lim_{x \to 1} f(x) = f(1) lim x → 1 f ( x ) = f ( 1 ) what are possible values of a a a and b b b ? NCERT textbook · Ex 12.1 Q29 Moderate Add #41 · Limits of Polynomials and Rational Functions
Let a 1 , a 2 , … , a n a_1, a_2, \ldots, a_n a 1 , a 2 , … , a n be fixed real numbers and define a function f ( x ) = ( x − a 1 ) ( x − a 2 ) … ( x − a n ) f(x) = (x - a_1)(x - a_2) \ldots (x - a_n) f ( x ) = ( x − a 1 ) ( x − a 2 ) … ( x − a n ) . What is lim x → a 1 f ( x ) \lim_{x \to a_1} f(x) lim x → a 1 f ( x ) ? For some a ≠ a 1 , a 2 , … , a n a \neq a_1, a_2, \ldots, a_n a = a 1 , a 2 , … , a n , compute lim x → a f ( x ) \lim_{x \to a} f(x) lim x → a f ( x ) . NCERT textbook · Ex 12.1 Q30 Moderate Add #42 · Limits and the Algebra of Limits
If f ( x ) = { ∣ x ∣ + 1 , x < 0 0 , x = 0 ∣ x ∣ − 1 , x > 0 f(x) = \begin{cases} |x| + 1, & x < 0 \\ 0, & x = 0 \\ |x| - 1, & x > 0 \end{cases} f ( x ) = ⎩ ⎨ ⎧ ∣ x ∣ + 1 , 0 , ∣ x ∣ − 1 , x < 0 x = 0 x > 0 For what value(s) of a a a does lim x → a f ( x ) \lim_{x \to a} f(x) lim x → a f ( x ) exists? NCERT textbook · Ex 12.1 Q31 Moderate Add #43 · Limits and the Algebra of Limits
If the function f ( x ) f(x) f ( x ) satisfies lim x → 1 f ( x ) − 2 x 2 − 1 = π \lim_{x \to 1} \frac{f(x) - 2}{x^2 - 1} = \pi lim x → 1 x 2 − 1 f ( x ) − 2 = π , evaluate lim x → 1 f ( x ) \lim_{x \to 1} f(x) lim x → 1 f ( x ) . NCERT textbook · Ex 12.1 Q32 Hard Add #44 · Limits and the Algebra of Limits
If f ( x ) = { m x 2 + n , x < 0 n x + m , 0 ≤ x ≤ 1 n x 3 + m , x > 1 f(x) = \begin{cases} mx^2 + n, & x < 0 \\ nx + m, & 0 \le x \le 1 \\ nx^3 + m, & x > 1 \end{cases} f ( x ) = ⎩ ⎨ ⎧ m x 2 + n , n x + m , n x 3 + m , x < 0 0 ≤ x ≤ 1 x > 1 For what integers m m m and n n n does both lim x → 0 f ( x ) \lim_{x \to 0} f(x) lim x → 0 f ( x ) and lim x → 1 f ( x ) \lim_{x \to 1} f(x) lim x → 1 f ( x ) exist? NCERT textbook · 12.2 Eg.5 Easy Add #45 · Derivatives from First Principles
Find the derivative at x = 2 x = 2 x = 2 of the function f ( x ) = 3 x f(x) = 3x f ( x ) = 3 x . NCERT textbook · 12.2 Eg.6 Moderate Add #46 · Derivatives from First Principles
Find the derivative of the function f ( x ) = 2 x 2 + 3 x − 5 f(x) = 2x^2 + 3x - 5 f ( x ) = 2 x 2 + 3 x − 5 at x = − 1 x = -1 x = − 1 . Also prove that f ′ ( 0 ) + 3 f ′ ( − 1 ) = 0 f'(0) + 3f'(-1) = 0 f ′ ( 0 ) + 3 f ′ ( − 1 ) = 0 . NCERT textbook · 12.2 Eg.7 Easy Add #47 · Derivatives from First Principles
Find the derivative of sin x \sin x sin x at x = 0 x = 0 x = 0 . NCERT textbook · 12.2 Eg.8 Easy Add #48 · Derivatives from First Principles
Find the derivative of f ( x ) = 3 f(x) = 3 f ( x ) = 3 at x = 0 x = 0 x = 0 and at x = 3 x = 3 x = 3 . NCERT textbook · 12.2 Eg.9 Easy Add #49 · Derivatives from First Principles
Find the derivative of f ( x ) = 10 x f(x) = 10x f ( x ) = 10 x . NCERT textbook · 12.2 Eg.10 Easy Add #50 · Derivatives from First Principles
Find the derivative of f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 . Showing the first 50 of 126.
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