Limits Questions with Solutions

21 questions7 easy13 medium1 hard

The limit of a function describes the value the function approaches as its input approaches a given point. Limits are the foundation of calculus, because continuity, derivatives and integrals are all defined using them. This topic covers left-hand and right-hand limits, the algebra of limits, and standard methods of evaluation: direct substitution, factorisation, rationalisation, and dividing by the highest power for limits at infinity. It also covers the standard limits, such as (sin x)/x -> 1 as x -> 0, (e^x - 1)/x -> 1, (a^x - 1)/x -> log a, and (1 + x)^(1/x) -> e. Most exam questions are 0/0 forms that reduce to one of these after simplification. A limit exists only when the left-hand and right-hand limits are equal, which is especially important for the modulus and greatest integer functions. Practise the limits questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
Find the value of $l im_{x \to \infty} \frac{x}{x + x + x}$
A. 0B. $2$ C. 1D. $3$
easyAsked in NIMCET 2026View solution ›
Q2.
The value of the limit $l im_{x \to 0} \frac{∣ x ∣ l o g _{e} ( 1 + ∣ s in 2 x ∣ )}{x ^{2} ( ∣ x ∣ + 3 )}$ is:
A. exists and is equal to $\frac{2}{3}$ B. exists and is equal to $\frac{1}{3}$ C. does not existD. exists and is equal to 0
mediumAsked in NIMCET 2026View solution ›
Q3.
Evaluate $l im_{x \to 0} \frac{x ^{2} + 2 cos x - 2}{x s in ^{3} x}$
A. 1/2B. 1/3C. 1/6D. 1/12
mediumAsked in NIMCET 2026View solution ›
Q4.
If $l im_{x \to 0} \frac{l o g ( 3 + x ) - l o g ( 3 - x )}{x} = K$ , then $K$ is
A. $\frac{2}{5}$ B. $\frac{2}{3}$ C. $\frac{1}{2}$ D. $\frac{5}{2}$
easyAsked in AMU MCA 2016View solution ›
Q5.
If $l im_{x \to 1} \frac{x ^{4} - 1}{x - 1} = l im_{x \to k} \frac{x ^{3} - k ^{3}}{x ^{2} - k ^{2}}$ , then find $k$ .
A. $\frac{8}{3}$ B. $\frac{4}{3}$ C. $\frac{2}{3}$ D. 1
easyAsked in NIMCET 2023View solution ›
Q6.
Let $f (x) = \frac{x ^{2} - 1}{∣ x ∣ - 1}$ . Then the value of $l im_{x \to - 1} f (x)$ is
A. -1B. 1C. 2D. 3
easyAsked in NIMCET 2023View solution ›
Q7.
If $f (x) = \frac{6 ^{x} - 3 ^{x} - 2 ^{x} + 1}{l o g _{e} 9 ( 1 - cos x )}$ , then $l im_{x \to 0} f (x)$ is
A. 2B. 3C. $l o g_{e} 2$ D. $l o g_{e} 3$
mediumAsked in NIMCET 2023View solution ›
Q8.
$l im_{x \to 0} [\frac{t an x - x}{x ^{2} t an x}]$ is equal to
A. 0B. 1C. 1/2D. 1/3
mediumAsked in NIMCET 2013View solution ›
Q9.
$l im_{x \to 0} \frac{x + s in x}{x - cos x}$ is equal to
A. $0$ B. $1$ C. $- 1$ D. none of these
easyAsked in NIMCET 2011View solution ›
Q10.
The integer $n$ for which $l im_{x \to 0} \frac{( cos x - 1 ) ( cos x - e ^{x} )}{x ^{n}}$ is finite and non-zero is
A. $1$ B. $2$ C. $3$ D. $4$
mediumAsked in NIMCET 2008View solution ›
Q11.
$l im_{x \to \infty} (\frac{x + 7}{x + 2})^{x + 5}$ is equal to
A. $e^{5}$ B. $e^{- 5}$ C. $e^{2}$ D. $e^{- 2}$
mediumAsked in NIMCET 2021View solution ›
Q12.
Which of the following is NOT true?
A. $l im_{x \to \infty} \frac{x}{e ^{x}} = 0$ B. $l im_{x \to 0^{+}} \frac{1}{x e ^{1/ x}} = 0$ C. $l im_{x \to 0^{+}} \frac{s in x}{1 + 2 x} = 0$ D. $l im_{x \to 0^{+}} \frac{cos x}{1 + 2 x} = 0$
mediumAsked in NIMCET 2022View solution ›
Q13.
If $f (x) = {\frac{s in [ x ]}{[ x ]}, [x] \neq = 00, [x] = 0$ , where [x] is the largest integer but not larger than x, then $l im_{x \to 0} f (x)$ is
A. -1B. 0C. 1D. Does not exist
mediumAsked in NIMCET 2014View solution ›
Q14.
The value of $l im_{x \to a} \frac{a + 2 x - 3 x}{3 a + x - 2 x}$ is
A. $\frac{2}{3}$ B. $\frac{2}{3}$ C. $\frac{3 3}{2}$ D. $\frac{2}{3 3}$
mediumAsked in NIMCET 2015View solution ›
Q15.
Let f(x) be a polynomial of degree four, having extreme values at x = 1 and x = 2. If $l im_{x \to 0} [1 + \frac{f ( x )}{x ^{2}}] = 3$ , then f(2) is
A. 0B. 4C. −8D. −4
hardAsked in NIMCET 2017View solution ›
Q16.
$l im_{x \to 0} \frac{x t an x}{1 - cos x}$ is
A. $\frac{1}{2}$ B. $- \frac{1}{2}$ C. −2D. 2
easyAsked in NIMCET 2017View solution ›
Q17.
If $l im_{x \to \infty} (1 + \frac{a}{x} + \frac{b}{x ^{2}})^{2 x} = e^{2}$ , then the values of a and b are:
A. $a = 1$ and $b = 2$ B. $a = 1$ , $b \in R$ C. $a \in R$ , $b = 2$ D. $a \in R$ , $b \in R$
mediumAsked in NIMCET 2018View solution ›
Q18.
$l im_{x \to 3} \frac{3 x - 3}{2 x - 4 - 2}$ is equal to
A. $3$ B. $\frac{3}{2}$ C. $\frac{1}{2 2}$ D. $\frac{1}{2}$
mediumAsked in NIMCET 2019View solution ›
Q19.
The value of the limit $l im_{x \to 0} (\frac{1 ^{x} + 2 ^{x} + 3 ^{x} + 4 ^{x}}{4})^{1/ x}$ is
A. 1B. $(3!)^{1/3!}$ C. $(3!)^{1/4}$ D. $(4!)^{1/4}$
mediumAsked in NIMCET 2024View solution ›
Q20.
The value of $l im_{x \to 0} \frac{e ^{x} - e ^{- x} - 2 x}{1 - cos x}$ is equal to
A. 2B. 1C. 0D. -1
mediumAsked in NIMCET 2024View solution ›

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