Chapter 11

The value of \(\lim _{n \rightarrow \infty} \frac{1}{n^{4}}\left[1\left(\sum_{k=1}^{n} k\right)+2\left(\sum_{k=1}^{n-1} k\right)+3\left(\sum_{k=1}^{n-2} k\right)+\ldots+n \cdot 1\right]\) will be (A) \(\frac{1}{24}\) (B) \(\frac{1}{12}\) (C) \(\frac{1}{6}\) (D) \(\frac{1}{3}\)

If \([x]\) denotes the integral part of \(x\), then \(\lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left(\sum_{k=1}^{n}\left[k^{2} x\right]\right)=\) (A) 0 (B) \(\frac{x}{2}\) (C) \(\frac{x}{3}\) (D) \(\frac{x}{6}\)

\(\lim _{n \rightarrow \infty}\left(\tan \theta+\frac{1}{2} \tan \frac{\theta}{2}+\frac{1}{2^{2}} \tan \frac{\theta}{2^{2}}+\ldots+\frac{1}{2^{n}} \tan \frac{\theta}{2^{n}}\right)=\) (A) \(\frac{1}{\theta}\) (B) \(\frac{1}{\theta}-2 \cot 2 \theta\) (C) \(2 \cot 2 \theta\) (D) None of these

\(\lim _{n \rightarrow 0} \frac{4^{3 n-2}-9^{n+1}}{8^{2 n-1}-9^{n-1}}=\) (A) \(\frac{1}{2}\) (B) 81 (C) Does not exist (D) None of these

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

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

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

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

\(\lim _{n \rightarrow \infty}(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \ldots\left(1+x^{2 n}\right),|x|<1\), is equal to (A) \(\frac{1}{x-1}\) (B) \(\frac{1}{1-x}\) (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 \(n\)

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

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

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) \(\infty\)

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 \(A=h \sqrt{2 h r-h^{2}}\), where \(h\) is the altitude from \(A\) to \(B C\), then \(\lim _{h \rightarrow 0^{+}} \frac{A}{P^{3}}\) is equal to \(\begin{array}{ll}\text { (A) } 128 r & \text { (B) } \frac{1}{128 r}\end{array}\) (C) \(\frac{1}{64 r}\) (D) None of these

\(\lim _{x \rightarrow \pi / 3} \frac{\cos \left(x+\frac{\pi}{6}\right)}{(1-2 \cos x)^{2 / 3}}=\) (A) 1 (B) \(-1\) (C) 0 (D) None of these

\(\lim _{x \rightarrow 0} \frac{\ln (2-\cos 2 x)}{\ln ^{2}(\sin 3 x+1)}\) is equal to (A) \(\frac{2}{9}\) (B) \(-\frac{2}{9}\) (C) 0 (D) None of these

\(\lim _{x \rightarrow 1 / \alpha} \frac{1-\cos \left(c x^{2}+b x+a\right)}{(1-x \alpha)^{2}}\), where \(\alpha\) is a root of \(a x^{2}+b x+c=0\), is equal to (A) \(\frac{b^{2}-4 a c}{2 \alpha^{2}}\) (B) \(\frac{b^{2}-4 a c}{\alpha^{2}}\) (C) \(\frac{4 a c-b^{2}}{2 \alpha^{2}}\) (D) None of these

\(\lim _{x \rightarrow 0} \frac{\sqrt{1-\sqrt{\cos x}}}{x}=\) (A) \(\frac{1}{2}\) (B) \(-\frac{1}{2}\) (C) Does not exist (D) None of these

\(\lim _{x \rightarrow 2} \frac{\sqrt{x+7}-3 \sqrt{2 x-3}}{\sqrt[3]{x+6}-2 \sqrt[3]{3 x-5}}=\) (A) \(\frac{17}{23}\) (B) \(\frac{34}{23}\) (C) 1 (D) None of these

\(\lim _{x \rightarrow 0} \frac{\left(2^{m}+x\right)^{1 / m}-\left(2^{n}+x\right)^{1 / n}}{x}\) is equal to (A) \(\frac{1}{m 2^{m}}-\frac{1}{n 2^{n}}\) (B) \(\frac{1}{m 2^{m}}+\frac{1}{n 2^{n}}\) (C) \(\frac{1}{m 2^{m-1}}-\frac{1}{n 2^{n-1}}\) (D) None of these

\(\lim _{x \rightarrow 4} \frac{(\cos \theta)^{x}-(\sin \theta)^{x}-\cos 2 \theta}{x-4}=\) (A) \(\cos ^{4} \theta \ln \cos \theta-\sin ^{4} \theta \ln \sin \theta\) (B) \(\cos ^{4} \theta \ln \cos \theta+\sin ^{4} \theta \ln \sin \theta\) (C) \(\cos ^{4} \theta \ln \sin \theta-\sin ^{4} \theta \ln \cos \theta\) (D) None of these

\(\lim _{x \rightarrow 0}\left(\frac{x-1+\cos x}{x}\right)^{1 / x}=\) (A) \(e^{1 / 2}\) (B) \(e^{-1 / 2}\) (C) \(e^{1 / 4}\) (D) None of these

\(\lim _{x \rightarrow \infty}\left[\frac{e}{(1+1 / x)^{x}}\right]^{x}=\) (A) \(e\) (B) \(e^{-1}\) (C) \(e^{1 / 2}\) (D) \(e^{-1 / 2}\)

\(\lim _{x \rightarrow 0}\left[\frac{a \sin x}{x}\right]+\left[\frac{b \tan x}{x}\right]\), where \(a, b\) are integers and [] denotes integral part, is equal to (A) \(a+b\) (B) \(a+b-1\) (C) \(a-b\) (D) \(a-b-1\)

\(\lim _{n \rightarrow \infty} \frac{[x]+[2 x]+[3 x]+\ldots+[n x]}{1+2+3+\ldots+n}=\) (A) \(x\) (B) \(2 x\) (C) 0 (D) None of these

\(\lim _{n \rightarrow \infty} n^{2}\left(x^{1 / n}-x^{1 / n+1}\right), x>0\) is equal to (A) 0 (B) \(e^{x}\) (C) \(\ln x\) (D) None of these

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

If \(y=x+\frac{\sqrt{x}}{x+\frac{\sqrt{x}}{x+\frac{\sqrt{x}}{\ldots \infty}}}\), then \(\lim _{x \rightarrow \infty} \frac{x}{y}\) is equal to (A) 1 (B) \(-1\) (C) 0 (D) None of these

\(\lim _{x \rightarrow 0} \frac{\cos x-(\cos x)^{\cos x}}{1-\cos x+\ln (\cos x)}=\) (A) 0 (B) 1 (C) 2 (D) None of these

The value of \(\lim _{x \rightarrow \pi / 4} \frac{(\tan x)^{\tan x}-\tan x}{\ln (\tan x)-\tan x+1}\) is (A) \(-2\) (B) 1 (C) 0 (D) None of these

\(\lim _{x \rightarrow \pi / 2}\left(1^{1 / \cos ^{2} x}+2^{1 \cos ^{2} x}+\ldots+n^{1 / \cos ^{2} x}\right)^{\cos ^{3} x}=\) (A) \(n\) (B) \(\frac{n(n+1)}{2}\) (C) \(n !\) (D) None of these

\(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \cot ^{-1}\left(r^{2}+\frac{3}{4}\right)=\) (A) 0 (B) \(\tan ^{-1} 2\) (C) \(\frac{\pi}{4}\) (D) None of these

The value of \(\lim _{x \rightarrow 5 \pi / 4}[\sin x+\cos x]\), where \([\cdot]\) denotes the greatest integer function, is (A) 2 (B) \(-2\) (C) 1 (D) \(-1\)

The value of \(\lim _{n \rightarrow \infty}\left[\sum_{r=1}^{n} \frac{1}{2^{r}}\right]\), where \([\cdot]\) denotes the greatest integer, is (A) 0 (B) 1 (C) \(-1\) (D) \(\frac{1}{2}\)

The value of \(\lim _{x \rightarrow \infty}|x|^{[\cos x]}\), where \([\cdot]\) denotes the greatest integer, is (A) 0 (B) 1 (C) \(-1\) (D) Does not exist

If \(a_{1}=1\) and \(a_{n}=n\left(1+a_{n-1}\right), \forall n \geq 2\), then \(\lim _{n \rightarrow \infty}\left(1+\frac{1}{a_{1}}\right)\left(1+\frac{1}{a_{2}}\right) \ldots\left(1+\frac{1}{a_{n}}\right)=\) (A) 0 (B) \(e\) (C) \(e^{2}\) (D) Does not exist

\(\lim _{n \rightarrow \infty} n^{-n^{2}}\left[(n+1)\left(n+\frac{1}{2}\right)\left(n+\frac{1}{2^{2}}\right) \ldots\left(n+\frac{1}{2^{n-1}}\right)\right]^{n}\) (A) \(e\) (B) \(e^{2}\) (C) \(e^{4}\) (D) None of these

If \(\lim _{x \rightarrow y} \frac{x^{y}-y^{x}}{x^{x}-y^{y}}=\frac{1-k}{1+k}\), then \(k=\) (A) \(\log y\) (B) \(e^{y}\) (C) \(y\) (D) None of these

\(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \cot ^{-1}\left(\frac{r^{3}-r+\frac{1}{r}}{2}\right)\) is equal to (A) 0 (B) \(\pi\) (C) \(\frac{\pi}{2}\) (D) None of these

\(\lim _{x \rightarrow 0}\left[\frac{x^{2}}{\sin x \tan x}\right]\), where \([\cdot]\) denotes the greatest integer function, is (A) 0 (B) 1 (C) 2 (D) Does not exist

\(\lim _{\theta \rightarrow 0} \frac{\cos ^{2}\left(1-\cos ^{2}\left(1-\cos ^{2}\left(1 \ldots \cos ^{2} \theta\right)\right)\right.}{\sin \left(\frac{\pi(\sqrt{\theta+4}-2)}{\theta}\right)}=\) (A) 1 (B) 0 (C) \(\sqrt{2}\) (D) \(-\sqrt{2}\)

\(\lim _{x \rightarrow 1} \frac{(\log (1+x)-\log 2)\left(3.4^{x-1}-3 x\right)}{\left\\{(7+x)^{1 / 3}-(1+3 x)^{1 / 2}\right\\} \sin \pi x}=\) (A) \(\frac{9}{\pi} \log \frac{4}{e}\) (B) \(\frac{3}{\pi} \log \frac{4}{e}\) (C) \(\frac{9}{\pi} \log \frac{2}{e}\) (D) None of these

\(\lim _{x \rightarrow 1} \frac{(1-x)\left(1-x^{2}\right) \ldots\left(1-x^{2 n}\right)}{\left[(1-x)\left(1-x^{2}\right) \ldots\left(1-x^{n}\right)\right]^{2}}=\) (A) \(n !\) (B) \(\frac{(2 n) !}{n !}\) (C) \(\frac{(2 n) !}{(n !)^{2}}\) (D) None of these

If \(\sum_{r=1}^{k} \cos ^{-1} \alpha r=\frac{k \pi}{2}\) for any \(k \geq 1\) and \(\theta=\sum_{r=1}^{k}(\alpha r)^{r}\), then \(\lim _{x \rightarrow \theta} \frac{\left(1+x^{2}\right)^{1 / 3}-(1-2 x)^{1 / 4}}{x+x^{2}}\) is (A) 1 (B) \(-1\) (C) \(\frac{1}{2}\) (D) \(-\frac{1}{2}\)

Given a real valued function \(f\) such that \(f(x)=\left\\{\begin{array}{cl}\frac{\tan ^{2}\\{x\\}}{x^{2}-[x]^{2}} & , x>0 \\\ 1 & , x=0 \\ \sqrt{\\{x\\} \cot \\{x\\}} & , x<0\end{array}\right.\) The value of \(\cot ^{-1}\left(\lim _{x \rightarrow 0} f(x)\right)^{2}\) is (A) 0 (B) 1 (C) \(-1\) (D) None of these

If \(\lim _{x \rightarrow 0} \frac{x^{a} \sin ^{b} x}{\sin \left(x^{c}\right)}, a, b, c \in R-\\{0\\}\) exists and has non-zero value, then (A) \(a, b, c\) are in A.P. (B) \(a, c, b\) are in A.P. (C) \(a, c, b\) are in G.P. (D) None of these

If \(\lim _{x \rightarrow 0} \frac{a x e^{x}-b \log (1+x)+c x e^{-x}}{x^{2} \sin x}=2\), then (A) \(a=3\) (B) \(b=12\) (C) \(c=9\) (C) \(a=-3\)

We know that if \(\lim _{x \rightarrow a} f(x)=l\) and \(\lim _{x \rightarrow a} g(x)=m(\neq 0)\), then $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)} $$ However, if \(\lim _{x \rightarrow a} g(x)=0=\lim _{x \rightarrow a} f(x)\), we cannot say anything definite about the existence of \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\). Though in some cases this limit exists. Any expression of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) is termed as an indeterminate form. Many other expressions like \(\infty-\infty, 1^{\infty}, \infty^{0}, 0^{\circ}, 0 \times \infty\) which can be reduced to the form \(\frac{0}{0}\) or \(-\infty\) are also called indeterminate forms. then If \(\frac{f(x)}{g(x)}\) is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} $$ wheref \(^{\prime}\) is derivative of \(f\). If \(\frac{f^{\prime}(x)}{g^{\prime}(x)}\), too, is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lim _{x \rightarrow a} \frac{f^{\prime \prime}(x)}{g^{\prime \prime}(x)}\) This can be continued till we finally arrive at a determinate result. If \(\lim _{x \rightarrow 0} \frac{\sin 2 x+a \sin x}{x^{3}}\) be finite, then the value of \(a\) and the limit are given by (A) \(-2,1\) (B) \(-2,-1\) (C) 2,1 (D) \(2,-1\)

We know that if \(\lim _{x \rightarrow a} f(x)=l\) and \(\lim _{x \rightarrow a} g(x)=m(\neq 0)\), then $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)} $$ However, if \(\lim _{x \rightarrow a} g(x)=0=\lim _{x \rightarrow a} f(x)\), we cannot say anything definite about the existence of \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\). Though in some cases this limit exists. Any expression of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) is termed as an indeterminate form. Many other expressions like \(\infty-\infty, 1^{\infty}, \infty^{0}, 0^{\circ}, 0 \times \infty\) which can be reduced to the form \(\frac{0}{0}\) or \(-\infty\) are also called indeterminate forms. then If \(\frac{f(x)}{g(x)}\) is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} $$ wheref \(^{\prime}\) is derivative of \(f\). If \(\frac{f^{\prime}(x)}{g^{\prime}(x)}\), too, is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lim _{x \rightarrow a} \frac{f^{\prime \prime}(x)}{g^{\prime \prime}(x)}\) This can be continued till we finally arrive at a determinate result. The value of \(\lim _{x \rightarrow 0} \sqrt{a^{2}-x^{2}} \cot \frac{\pi}{2} \sqrt{\frac{a-x}{a+x}}\) is (A) \(\frac{2 a}{\pi}\) (B) \(-\frac{2 a}{\pi}\) (C) \(\frac{4 a}{\pi}\) (D) \(-\frac{4 a}{\pi}\)

For a function \(f\), let \(\lim _{x \rightarrow a} f(x) \neq 1\) but \(f(x)\) is $$ \begin{aligned} &\qquad \lim _{x \rightarrow a}\\{f(x)\\}^{g(x)}, \\ &\text { we write }\\{f(x)\\}^{g(\tau)}=e^{\log _{t}(f(x)\\}^{r * *}} \\ &\Rightarrow \quad \lim _{x \rightarrow a}\\{f(x)\\}^{g(x)}=e^{\lim g(x) \log f(x)} \end{aligned} $$ In case, \(\lim _{x \rightarrow a} f(x)=1\) and \(\lim _{x \rightarrow a} g(x)=\infty\), then $$ \begin{aligned} \lim _{x \rightarrow a}\\{f(x)\\}^{g(x)} &=\lim _{\cdots}(1+f(x)-1)^{g(x)} \\ &=e^{\lim (f(x)-1) g(x)} \end{aligned} $$ If \(\alpha\) and \(\beta\) be the roots of \(a x^{2}+b x+c=0\), then \(\lim _{x \rightarrow \alpha}\left(1+a x^{2}+b x+c\right)^{1(x-\alpha)}\) is (A) \(\log |a(\alpha-\beta)|\) (B) \(e^{a(\alpha-\beta)}\) (C) \(e^{a(\beta-\alpha)}\) (D) None of these