JEE Mathematics Limits Indeterminate Forms MCQs Set 01

Choose the most appropriate option (a, b, c or d).

Question. \(\lim _{x \rightarrow 1} \frac{(\sqrt{x}-1)(2 x-3)}{2 x^{2}+x-3}\) is equal to
(a) \(\frac{1}{10}\)
(b) \(-\frac{1}{10}\)
(c) 1
(d) None of the options
Answer: (b) \(-\frac{1}{10}\)

Question. \(\lim _{x \rightarrow 2} \frac{\sqrt{x-2}+\sqrt{x}-\sqrt{2}}{\sqrt{x^{2}-4}}\) is equal to
(a) \(\frac{1}{2}\)
(b) 1
(c) 2
(d) None of the options
Answer: (a) \(\frac{1}{2}\)

Question. \(\lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x}}-\sqrt{x})\) is equal to
(a) 1
(b) 0
(c) \(\frac{1}{2}\)
(d) None of the options
Answer: (c) \(\frac{1}{2}\)

Question. \(\lim _{n \rightarrow \infty} \frac{a^{n}+b^{n}}{a^{n}-b^{n}}\), where a > b > 1, is equal to
(a) -1
(b) 1
(c) 0
(d) None of the options
Answer: (b) 1

Question. \(\lim _{x \rightarrow 0} \frac{3^{x}-1}{\sqrt{x+1}-1}\) is equal to
(a) \(\log _{e} 9\)
(b) \(\log _{e} 3\)
(c) 0
(d) 1
Answer: (a) \(\log _{e} 9\)

Question. \(\lim _{n \rightarrow \infty} \frac{4^{1 / n}-1}{3^{1 / n}-1}\) is equal to
(a) \(\log _{4} 3\)
(b) 1
(c) \(\log _{3} 4\)
(d) None of the options
Answer: (c) \(\log _{3} 4\)

Question. \(\lim _{x \rightarrow \infty} \frac{\cos x+\sin x}{x^{2}}\) is equal to
(a) 1
(b) 0
(c) \(\infty\)
(d) None of the options
Answer: (b) 0

Question. \(\lim _{x \rightarrow 0} \frac{3^{x}-2^{x}}{4^{x}-3^{x}}\) is equal to
(a) 1
(b) -1
(c) 0
(d) None of the options
Answer: (d) None of the options

Question. \(\lim _{h \rightarrow 0}\left\{\frac{1}{h \cdot \sqrt[3]{8+h}}-\frac{1}{2 h}\right\}\) is equal to
(a) \(\frac{1}{12}\)
(b) \(-\frac{4}{3}\)
(c) \(-\frac{16}{3}\)
(d) \(-\frac{1}{48}\)
Answer: (d) \(-\frac{1}{48}\)

Question. \(\lim _{n \rightarrow \infty} \frac{n^{p} \sin ^{2}(n !)}{n+1}, 0
(a) 0
(b) \(\infty\)
(c) 1
(d) None of the options
Answer: (a) 0

Question. \(\lim _{x \rightarrow 0} \frac{x \sqrt{y^{2}-(y-x)^{2}}}{\left(\sqrt{8 x y-4 x^{2}}+\sqrt{8 x y}\right)^{3}}\) is equal to
(a) \(\frac{1}{4}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{2 \sqrt{2}}\)
(d) None of the options
Answer: (a) \(\frac{1}{4}\)

Question. \(\lim _{x \rightarrow 0} \frac{\left(1+x+x^{2}\right)-e^{x}}{x^{2}}\) is equal to
(a) 1
(b) 0
(c) \(\frac{1}{2}\)
(d) None of the options
Answer: (c) \(\frac{1}{2}\)

Question. \(\lim _{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^{2}}\) is equal to
(a) 2
(b) -2
(c) \(\frac{1}{2}\)
(d) \(-\frac{1}{2}\)
Answer: (c) \(\frac{1}{2}\)

Question. \(\lim _{x \rightarrow 0}\left\{\frac{\log _{e}(1+x)}{x^{2}}+\frac{x-1}{x}\right\}\) is equal to
(a) \(\frac{1}{2}\)
(b) \(-\frac{1}{2}\)
(c) 1
(d) None of the options
Answer: (a) \(\frac{1}{2}\)

Question. \(\lim _{n \rightarrow \infty}(0.2)^{\log _{\sqrt{5}}(1 / 4+1 / 8+1 / 16+\dots \text{ to } n \text{ terms })}\) is equal to
(a) 2
(b) 4
(c) 8
(d) 0
Answer: (b) 4

Question. Let \(s_{n}=\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+\dots\) to n terms. Then \(\lim _{n \rightarrow \infty} s_{n}\) is equal to
(a) \(\frac{1}{3}\)
(b) 3
(c) \(\frac{1}{4}\)
(d) \(\infty\)
Answer: (a) \(\frac{1}{3}\)

Question. Let the rth term, \(t_{r}\), of a series is given by \(t_{r}=\frac{r}{1+r^{2}+r^{4}}\). Then \(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} t_{r}\) is
(a) \(\frac{1}{4}\)
(b) 1
(c) \(\frac{1}{2}\)
(d) None of the options
Answer: (c) \(\frac{1}{2}\)

Question. Let \(a=\min \left\{x^{2}+2 x+3, x \in R\right\}\) and \(b=\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta^{2}}\). The value of \(\sum_{r=0}^{n} a^{r} \cdot b^{n-r}\) is
(a) \(\frac{2^{n+1}-1}{3 \cdot 2^{n}}\)
(b) \(\frac{2^{n+1}+1}{3 \cdot 2^{n}}\)
(c) \(\frac{4^{n+1}-1}{3 \cdot 2^{n}}\)
(d) None of the options
Answer: (c) \(\frac{4^{n+1}-1}{3 \cdot 2^{n}}\)

Question. \(\lim _{x \rightarrow 1} \frac{\sum_{r=1}^{n} x^{r}-n}{x-1}\) is equal to
(a) \(\frac{n}{2}\)
(b) \(\frac{n(n+1)}{2}\)
(c) 1
(d) None of the options
Answer: (b) \(\frac{n(n+1)}{2}\)

Question. \(\lim _{x \rightarrow \infty} \frac{x-\tan x}{x+\tan x}\) is equal to
(a) 0
(b) 1
(c) \(\frac{1}{2}\)
(d) None of the options
Answer: (a) 0

Question. \(\lim _{x \rightarrow 0} \sqrt{\frac{x-\sin x}{x+\sin ^{2} x}}\) is equal to
(a) 1
(b) 0
(c) \(\infty\)
(d) None of the options
Answer: (b) 0

Question. \(\lim _{x \rightarrow 0} \frac{\log _{e} \cos x}{x^{2}}\) is equal to
(a) \(-\frac{1}{2}\)
(b) \(\frac{1}{2}\)
(c) 0
(d) None of the options
Answer: (a) \(-\frac{1}{2}\)

Question. \(\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{x^{2}}\) is equal to
(a) \(\frac{1}{2}\)
(b) \(-\frac{1}{2}\)
(c) 0
(d) \(\infty\)
Answer: (c) 0

Question. \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x\left(2^{x}-1\right)}\) is equal to
(a) \(\frac{1}{2} \log _{2} e\)
(b) \(\frac{1}{2} \log _{e} 2\)
(c) 1
(d) None of the options
Answer: (a) \(\frac{1}{2} \log _{2} e\)