Chapter 2

\(f(x)=[x-1]+|x-1| ; x \neq 1\) where [.] denotes greatest integer function then \(f(1-0)\) and \(f(1\) \(+0\) ) respectively will be (a) \(-1,0\) (b) \(0,-1\) (a) \(-1,-1\) (b) 0,0

\(\lim _{x \rightarrow 2^{+}}\left(\frac{[x]^{3}}{3}-\left[\frac{x}{3}\right]^{3}\right)=\) (a) 0 (b) \(\frac{64}{27}\) (c) \(\frac{8}{3}\) (d) None of these

If \(f(x)=\left\\{\begin{array}{ll}x, & x<0 \\ 1, & x=0 \\ x^{2}, & x>0\end{array}\right.\), then \(\lim _{x \rightarrow 0} f(x)=\) (a) Is 1 (b) Is zero (c) Does not exist (d) None of these

\(\lim _{\alpha \rightarrow \beta}\left[\frac{\sin ^{2} \alpha-\sin ^{2} \beta}{\alpha^{2}-\beta^{2}}\right]=\) (a) 0 (b) 1 (c) \(\frac{\sin \beta}{\beta}\) (d) \(\frac{\sin 2 \beta}{2 \beta}\)

\(\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^{2}+x-3}=\) (a) \(-\frac{1}{10}\) (b) \(\frac{1}{10}\) (c) \(-\frac{1}{8}\) (d) None of these

If \(f(x)=\left\\{\begin{array}{cc}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.\), then \(\lim _{x \rightarrow 0} f(x)=\) (a) 1 (b) 0 (c) \(-1\) (d) None of these

Let the function \(f\) be defined by the equation \(f(x)=\left\\{\begin{array}{lll}3 x & \text { if } & 0 \leq x \leq 1 \\ 5-3 x & \text { if } & 1

\(\lim _{n \rightarrow \infty}\left[\frac{1}{1-n^{2}}+\frac{2}{1-n^{2}}+\ldots .+\frac{n}{1-n^{2}}\right]\) is equal to (a) 0 (b) \(-1 / 2\) (c) \(1 / 2\) (d) None of these

\(\lim _{n \rightarrow \infty}\left[\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots .+\frac{1}{(2 n+1)(2 n+3)}\right]\) is equal to (a) 0 (b) \(1 / 2\) (c) \(1 / 9\) (d) 2

\(\lim _{x \rightarrow \infty} \frac{(2 x+1)^{40}(4 x-1)^{5}}{(2 x+3)^{45}}=\) (a) 16 (b) 24 (c) 32 (d) 8

The value of \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+b x+4}{x^{2}+a x+5}\right)\) is (a) \(b / a\) (b) 1 (c) 0 (d) \(4 / 5\)

\(\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+8 x+3}-\sqrt{x^{2}+4 x+3}\right)\) is equal to (a) 0 (b) \(\infty\) (c) 2 (d) \(1 / 2\)

If \(x_{n}=\frac{1-2+3-4+5-6+\ldots \ldots \ldots 2 n}{\sqrt{n^{2}+1}+\sqrt{4 n^{2}-1}}\), then \(\lim _{n \rightarrow \infty} x_{n}\) is equal to (a) \(1 / 3\) (b) \(-1 / 3\) (c) \(2 / 3\) (d) \(-2 / 3\)

\(\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+a^{2}}-\sqrt{x^{2}+b^{2}}}{\sqrt{x^{2}+c^{2}}-\sqrt{x^{2}+d^{2}}}=\) (a) \(\frac{a^{2}-b^{2}}{c^{2}-d^{2}}\) (b) \(\frac{a^{2}+b^{2}}{c^{2}-d^{2}}\) (c) \(\frac{a^{2}+b^{2}}{c^{2}+d^{2}}\) (d) None of these

\(\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}}=\) (a) \(1 / 2\) (b) \(-1 / 2\) (c) \(2 / 3\) (d) None of these

\(\lim _{x \rightarrow 0} \frac{e^{\tan x}-e^{x}}{\tan x-x}=\) (a) 1 (b) \(e\) (c) \(e^{-1}\) (d) 0

\(\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{x^{3}}\) is equal to (a) 0 (b) 1 (c) \(-1\) (d) \(1 / 2\)

Values of constant \(a, b\) and \(c\) so that \(\lim _{x \rightarrow 0} \frac{x a e^{x}-b \log (1+x)+c x e^{-x}}{x^{2} \sin x}=2\), then \(a=\), \(b=, c=\) (a) \(3,12,9\) (b) \(9,6,9\) (c) \(5,10,20\) (d) None of these

\(\lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^{4}}=\) (a) \(1 / 8\) (b) \(1 / 2\) (c) \(1 / 4\) (d) None of these

The value of \(\lim _{x \rightarrow 7}\left(\frac{2-\sqrt{x-3}}{x^{2}-49}\right)\) is (a) \(2 / 9\) (b) \(-2 / 49\) (c) \(-1 / 56\) (d) \(-1 / 59\)

\(\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2} x\right)}{x^{2}}=\) (a) \(-\pi\) (b) \(\pi\) (c) \(\pi / 2\) (d) 1

\(\lim _{x \rightarrow \infty} \frac{\log x^{n}-[x]}{[x]}, n \in N,([x]\) denotes greatest integer less than or equal to \(x\) ) (a) Has value \(-1\) (b) Has value 0 (c) Has value 1 (d) Does not exist

If \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\), then (a) \(a=1\) and \(b=1\) (b) \(a=1\) and \(b=-1\) (c) \(a=1\) and \(b=-2\) (d) \(a=1\) and \(b=2\)

\(\lim _{x \rightarrow 0} \frac{a^{\sin x}-1}{b^{\sin x}-1}=\) (a) \(\frac{a}{b}\) (b) \(\frac{b}{a}\) (c) \(\frac{\log a}{\log b}\) (d) \(\frac{\log b}{\log a}\)

6\. \(\operatorname{Lt}_{x \rightarrow 1} \frac{x^{n}+x^{n-1}+x^{n-2}+\ldots \ldots+x^{2}+x-n}{x-1}\) is equal to (a) \(\frac{n(n+1)}{2}\) (b) \(\frac{n(n-1)}{2}\) (c) \(\frac{n(n+2)}{2}\) (d) None of these

\(\lim _{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)\) is equal to (a) 2 (b) \(1 / 2\) (c) \(\infty\) (d) 0

What is the value of \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}\) ? (a) 1 (b) 0 (c) \(\infty\) (d) \(-1\)

If \(\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{(x-1)^{2}}=2\), then \((a, b, c)\) is (a) \((2,-4,2)\) (b) \((2,4,2)\) (c) \((2,4,-2)\) (d) \((2,-4,-2)\) (e) \((-2,4,2)\)