Chapter 11

The value of \(\lim _{x \rightarrow 0}\left(\left[\frac{100 x}{\sin x}\right]+\left[\frac{99 \sin x}{x}\right]\right)\), where \([\cdot]\) represents greatest integer function, is (A) 199 (B) 198 (C) 0 (D) None of these

If \(f(x)=\sin x, \quad x \neq n \pi\) \(=2, \quad x=n \pi\) where \(n \in Z\) and \(g(x)=x^{2}+1, \quad x \neq 2\) \(=3\), \(x=2\). then \(\lim _{x \rightarrow 0} g[f(x)]\) is (A) 1 (B) 0 (C) 3 (D) Does not exist

The value of \(\lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})\) is (A) \(\frac{1}{2}\) 1 (C) 0 (D) None of these

The value of \(\lim _{x \rightarrow \infty} x\left[\tan ^{-1} \frac{x+1}{x+2}-\frac{\pi}{4}\right]\) is (A) \(\frac{1}{2}\) (B) \(-\frac{1}{2}\) (C) 1 (D) \(-1\)

\(\lim _{n \rightarrow \infty} \cos \left(\pi \sqrt{n^{2}+n}\right), n \in Z\) is equal to (A) 0 (B) 1 (C) \(-1\) (D) None of these

\(\lim _{n \rightarrow \infty} \frac{n^{k} \sin ^{2}(n !)}{n+2} 0

\(\lim _{x \rightarrow 1} \frac{\sqrt{1-\cos 2(x-1)}}{x-1}\) (A) exists and it equals \(\sqrt{2}\) (B) exists and it equals \(-\sqrt{2}\) (C) Does not exist because \((x-1) \rightarrow 0\) (D) Does not exist because left hand limit is not equal to right hand limit

The value of \(\lim _{x \rightarrow \infty} \frac{x^{5}}{5^{x}}\) is (A) 1 (B) \(-1\) (C) 0 (D) None of these

The value of \(\lim _{x \rightarrow \frac{\pi}{4}} \frac{2 \sqrt{2}-(\cos x+\sin x)^{3}}{1-\sin 2 x}\) is (A) \(\frac{3}{\sqrt{2}}\) (B) \(\frac{\sqrt{2}}{3}\) (C) \(\frac{1}{\sqrt{2}}\) (D) \(\sqrt{2}\)

The value of \(\lim _{h \rightarrow 0} \frac{\ln (1+2 h)-2 \ln (1+h)}{h^{2}}\) is (A) 1 (B) \(-1\) (C) 0 (D) None of these

The value of \(\lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{e^{1 / n}}{n}+\frac{e^{2 / n}}{n}+\ldots+\frac{e^{(n-1) / n}}{n}\right)\) is (A) 1 (B) 0 (C) \(e-1\) (D) \(e+1\)

$$ \begin{aligned} &\lim _{x \rightarrow 1} \frac{x \sin (x-[x])}{x-1}, \text { where }[\cdot] \text { denotes the greatest }\\\ &\text { integer function, is equal to } \end{aligned} $$ (A) 1 (B) \(-1\) (C) \(\infty\) (D) Does not exist

If \(f(x)=\int \frac{2 \sin x-\sin 2 x}{x^{3}} d x, x \neq 0\), then \(\lim _{x \rightarrow 0} f(x)\) is (A) 0 (B) \(\infty\) (C) \(-1\) (D) 1

\(\lim _{x \rightarrow \pi / 2} \frac{\left[\frac{x}{2}\right]}{\ln (\sin x)}\) (where [.] denotes the greatest integer function) (A) Does not exist (B) equals 1 (C) equals 0 (D) equals \(-1\)

\(\lim _{x \rightarrow 0}\left[\frac{\sin ([x-3])}{[x-3]}\right]\), where \([\cdot]\) represents greatest integer function, is (A) 0 (B) 1 (C) Does not exist (D) \(\sin 1\)

The values of constants \(a\) and \(b\) so that $$ \lim _{x \rightarrow \infty}\left(\frac{x^{2}+1}{x+1}-a x-b\right)=0 \text { are } $$ (A) \(a=1, b=-1\) (B) \(a=-1, b=1\) (C) \(a=0, b=0\) (D) \(a=2, b=-1\)

\(\lim _{n \rightarrow \infty}\left(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots+\frac{1}{n(n+1)}\right)\) is equal to (A) 1 (B) \(-1\) (C) 0 (D) None of these

\(\lim _{x \rightarrow \infty} \frac{(\log x)^{2}}{x^{n}}, n>0\) is equal to (A) 1 (B) 0 (C) \(-1\) (D) \(\infty\)

If the \(r\) th term, \(t_{r}\), of a series is given by \(t_{r}=\frac{r}{r^{4}+r^{2}+1}\), then \(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} t_{r}\) is (A) 1 (B) \(\frac{1}{2}\) (C) \(\frac{1}{3}\) (D) None of these

\(\lim _{x \rightarrow n}(-1)^{[x]}\), where \([x]\) denotes the greatest integer less than or equal to \(x\), is equal to (A) \((-1)^{n}\) (B) \((-1)^{n-1}\) (C) 0 (D) Does not exist

\(\lim _{x \rightarrow 1 \atop y \rightarrow 0} \frac{y^{3}}{x^{3}-y^{2}-1}\) as \((x, y) \rightarrow(1,0)\) along the line \(y=\) \(x-1\) is given by (A) 1 (B) \(\infty\) (C) 0 (D) None of these

\(\lim _{n \rightarrow \infty} \frac{1-2+3-4+5-6+\ldots-2 n}{\sqrt{n^{2}+1}+\sqrt{4 n^{2}-1}}\) is equal to (A) \(\frac{1}{3}\) (B) \(-\frac{1}{3}\) (C) \(-\frac{1}{5}\) (D) None of these

The value of \(\lim _{x \rightarrow-\infty}\left[\frac{x^{4} \sin (1 / x)+x^{2}}{1+|x|^{3}}\right]\) is (A) 1 (B) \(-1\) (C) 0 (D) \(\infty\)

\(\lim _{x \rightarrow 2} \frac{2^{x}+2^{3-x}-6}{2^{-x / 2}-2^{1-x}}\) is equal to (A) 8 (B) 4 (C) 2 (D) None of these

\(\lim _{x \rightarrow 0} \frac{8}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{4}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\) is equal to (A) \(\frac{1}{16}\) (B) \(-\frac{1}{16}\) (C) \(\frac{1}{32}\) (D) \(-\frac{1}{32}\)

\(\lim _{n \rightarrow \infty}\left[\log _{n-1}(n) \cdot \log _{n}(n+1) \cdot \log _{n+1}(n+2) \ldots \log _{n^{i}-1}\left(n^{k}\right)\right]\) is equal to (A) \(\infty\) (B) \(n\) (C) \(\underline{k}\) (D) None of these

\(\lim _{n \rightarrow \infty}\left[\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots+\frac{1}{(2 n+1)(2 n+3)}\right]\) is equal to (A) 1 (B) \(\frac{1}{2}\) (C) \(-\frac{1}{2}\) (D) None of these

The value of \(\lim _{x \rightarrow \infty}\left[\frac{1^{1 / x}+2^{1 / x}+3^{1 / x}+\ldots+n^{1 / x}}{n}\right]^{n x}\) is (A) \(n !\) (B) \(n\) (C) \((n-1) !\) (D) 0

$$ \begin{aligned} &\lim _{n \rightarrow \infty}(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \ldots\left(1+x^{2 n}\right),|x|<1, \text { is }\\\ &\text { equal to } \end{aligned} $$ (A) \(\frac{1}{x-1}\) (B) \(\frac{1}{x-1}\) (C) \(1-x\) (D) \(x-1\)

\(\lim _{x \rightarrow \infty} \frac{x^{n}}{e^{x}}=0,(n\) integer \()\), for (A) no value of \(n\) (B) all values of \(n\) (C) only negative values of \(n\) (D) only positive values of \(n\)

The value of \(\lim _{x \rightarrow 1} \frac{x^{n}+x^{n-1}+x^{n-2}+\ldots+x^{2}+x-n}{x-1}\) is (A) \(\frac{n(n+1)}{2}\) (B) 0 (C) 1 (D) \(n\)

If \(t_{r}=\frac{1^{2}+2^{2}+3^{2}+\ldots+r^{2}}{1^{3}+2^{3}+3^{3}+\ldots+r^{3}}\) and \(S_{n}=\sum_{r=1}^{n}(-1)^{r} \cdot t_{r}\), then \(\lim _{n \rightarrow \infty} \mathrm{S}_{n}\) is given by (A) \(\frac{2}{3}\) (B) \(-\frac{2}{3}\) (C) \(\frac{1}{3}\) (D) \(-\frac{1}{3}\)

If \(\lim _{x \rightarrow 0} \frac{\left(1+a^{3}\right)+8 e^{1 / x}}{1+\left(1-b^{3}\right) e^{1 / x}}=2\), then (A) \(a=1, b=(-3)^{1 / 3}\) (B) \(a=1, b=3^{1 / 3}\) (C) \(a=-1, b=-(3)^{1 / 3}\) (D) None of these

If \(a=\min \left\\{x^{2}+4 x+5, x \in R\right\\}\) and \(b=\lim _{\theta \rightarrow 0} \frac{1-\cos 2 \theta}{\theta^{2}}\), then the value of \(\sum_{r=0}^{n} a^{r} \cdot b^{n-r}\) is (A) \(\frac{2^{n+1}-1}{4 \cdot 2^{n}}\) (B) \(2^{n+1}-1\) (C) \(\frac{2^{n+1}-1}{3 \cdot 2^{n}}\) (D) None of these

\(\lim _{n \rightarrow \infty} \frac{1 \cdot 2+2 \cdot 3+3 \cdot 4+\ldots+n(n+1)}{n^{3}}\) is equal to (A) 1 (B) \(-1\) (C) \(\frac{1}{3}\) (D) None of these

\(\lim _{x \rightarrow 0} \frac{\log \left(1+x+x^{2}\right)+\log \left(1-x+x^{2}\right)}{\sec x-\cos x}\) is equal to \(\mathbf{4 8} .\) (A) 1 (B) \(-1\) (C) 0 (D) \(\infty\)

\(\lim _{x \rightarrow e} \frac{\ln x-1}{|x-e|}\) is equal to (A) \(\frac{1}{e}\) (B) \(-\frac{1}{e}\) (C) \(e\) (D) Does not exist

If \(x_{1}=3\) and \(x_{n+1}=\sqrt{2+x_{n}}, n \geq 1\), then \(\lim _{n \rightarrow \infty} x_{n}\) is equal to (A) \(-1\) (B) 2 (C) \(\sqrt{5}\) (D) 3

The value of \(\lim _{x \rightarrow \infty} \frac{3^{x+1}-5^{x+1}}{3^{x}-5^{x}}\) is (A) 5 (B) \(\frac{1}{5}\) (C) \(-5\) (D) None of these

\(\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{\frac{n-1}{n}}\right)\) is equal to (A) \(e\) (B) \(-e\) (C) \(e-1\) (D) \(1-e\)

\(\lim _{x \rightarrow \infty} \sqrt{\frac{x+\sin x}{x-\cos x}}=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

If \(S_{n}=\sum_{i=1}^{n} a_{i}\) and \(\lim _{n \rightarrow \infty} a_{n}=a\), then \(\lim _{n \rightarrow \infty} \frac{S_{n+1}-S_{n}}{\sqrt{\sum_{i=1}^{n} i}}\) is (A) 0 (B) \(a\) (C) \(\sqrt{2} a\) (D) None of these

The value of \(\lim _{n \rightarrow \infty}\left[\sqrt[3]{n^{2}-n^{3}}+n\right]\) is (A) \(\frac{1}{3}\) (B) \(-\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) \(-\frac{2}{3}\)

The value of \(\lim _{n \rightarrow \infty} \frac{\sqrt[4]{n^{5}+2}-\sqrt[3]{n^{2}+1}}{\sqrt[5]{n^{4}+2}-\sqrt[2]{n^{3}+1}}\) is (A) 1 (B) 0 (C) \(-1\) (D) \(\infty\)

The integer \(n\) for which \(\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}}\) is a finite non-zero number, is (A) 1 (B) 2 (C) 3 (D) 4

The value of \(\lim _{x \rightarrow \infty} \frac{2 \sqrt{x}+3 \sqrt[3]{x}+5 \sqrt[5]{x}}{\sqrt{3 x-2}+\sqrt[3]{2 x-3}}\) is (A) \(\frac{2}{\sqrt{3}}\) (B) \(\sqrt{3}\) (C) \(\frac{1}{\sqrt{3}}\) (D) None of these

$$ \lim _{x \rightarrow 0} \frac{x \sqrt[3]{z^{2}-(z-x)^{2}}}{\left(\sqrt[3]{8 x z-4 x^{2}}+\sqrt[3]{8 x z}\right)^{4}} \text { is equal to } $$ (A) \(\frac{z}{2^{11 / 3}}\) (B) \(\frac{1}{2^{23 / 3} \cdot z}\) (C) \(2^{21 / 3} z\) (D) None of these

In a circle of radius \(r\), an isosceles triangle \(A B C\) is inscribed with \(A B=A C\). If the \(\Delta A B C\) has perimeter \(P=\) \(2\left[\sqrt{2 h r-h^{2}}+\sqrt{2 h r}\right]\) and area \(A=h \sqrt{2 h r-h^{2}}\), \(57 .\) where \(h\) is the altitude from \(A\) to \(B C\), then \(\lim _{h \rightarrow 0^{-}} \frac{A}{P^{3}}\) is equal to (A) \(128 r\) (B) \(\frac{1}{128 r}\) (C) \(\frac{1}{64 r}\) (D) None of these

\(\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \\{2(x-2)\\}}}{x-2}\right)\) (A) equals \(\frac{1}{\sqrt{2}}\) (B) Does not exist (C) equals \(\frac{1}{\sqrt{2}}\) (D) equals \(-\sqrt{2}\)

\(\lim _{n \rightarrow \infty}\left(\cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \ldots \cos \frac{x}{2^{n}}\right)=\) (A) \(\frac{x}{\sin x}\) (B) \(\frac{\sin x}{x}\) (C) 0 (D) None of these