Chapter 11

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 \(\infty-\infty, 1^{\infty}, \infty^{0}, 0^{0}, 0 \times \infty\) which can be reduced to the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) are also called indeterminate forms. If \(\frac{f(x)}{g(x)}\) is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} $$ where \(f^{\prime}\) 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 \(\infty-\infty, 1^{\infty}, \infty^{0}, 0^{0}, 0 \times \infty\) which can be reduced to the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) are also called indeterminate forms. If \(\frac{f(x)}{g(x)}\) is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} $$ where \(f^{\prime}\) 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) \(-\frac{4 a}{\pi}\)

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 \(\infty-\infty, 1^{\infty}, \infty^{0}, 0^{0}, 0 \times \infty\) which can be reduced to the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) are also called indeterminate forms. If \(\frac{f(x)}{g(x)}\) is indeterminate at \(x=a\) of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then $$ \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} $$ where \(f^{\prime}\) 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. \(\lim _{x \rightarrow 0}|x|^{\sin x}\) equals (A) 0 (B) 1 (C) \(-1\) (D) None of these

Let \(f, g\) and \(h\) be real valued functions defined on an interval \(I \subseteq R\) except possibly for some point \(c\) such that $$ \lim _{x \rightarrow c} f(x)=l=\lim _{x \rightarrow c} h(x) $$ and, \(f(x) \leq g(x) \leq h(x), \forall x \in I\). Then, \(\lim _{x \rightarrow c} g(x)=l\). \(\lim _{n \rightarrow \infty} \frac{\\{x\\}+\\{2 x\\}+\\{3 x\\}+\ldots+\\{n x\\}}{n^{2}}\) where \(\\{x\\}=x-[x]\) denotes the fractional part of \(x\), is (A) 1 (B) 0 (C) \(\frac{1}{2}\) (D) None of these

Let \(f, g\) and \(h\) be real valued functions defined on an interval \(I \subseteq R\) except possibly for some point \(c\) such that $$ \lim _{x \rightarrow c} f(x)=l=\lim _{x \rightarrow c} h(x) $$ and, \(f(x) \leq g(x) \leq h(x), \forall x \in I\). Then, \(\lim _{x \rightarrow c} g(x)=l\). \(\lim _{x \rightarrow 0^{+}}\left(\lim _{n \rightarrow \infty} \frac{\left[1^{2} x^{x}\right]+\left[2^{2} x^{x}\right]+\ldots+\left[n^{2} x^{x}\right]}{n^{3}}\right)\), where \([\cdot]\) denotes the greatest integer function, is equal to (A) \(-\frac{1}{3}\) (B) \(\frac{1}{3}\) (C) 0 (D) None of these

Column-I (I) \(\lim _{n \rightarrow \infty}\left[\sqrt[3]{n^{2}-n^{3}}+n\right]\) (II) \(\lim _{x \rightarrow 1} \frac{\sqrt[3]{x^{2}}-2 \sqrt[3]{x}+1}{(x-1)^{2}}\) (III) \(\lim _{n \rightarrow \infty} \prod_{r=3}^{n}\left(\frac{r^{3}-1}{r^{3}+1}\right)\) (IV) \(\lim _{n \rightarrow \infty}\left(\cos \frac{x}{n}\right)^{n}\) Column-II (A) \(\frac{1}{9}\) (B) \(\frac{6}{7}\) (C) \(\frac{1}{3}\) (D) 1

Column-I (I) \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\ln \sin x}\) (II) \(\lim _{n \rightarrow \infty}\left\\{\frac{7}{10}+\frac{29}{10^{2}}+\frac{133}{10^{3}}+\ldots\right.\) \(\left.+\frac{5^{n}+2^{n}}{10^{n}}\right\\}\) (III) \(\lim _{x \rightarrow \infty} x\left[\tan ^{-1} \frac{x+1}{x+2}-\frac{\pi}{4}\right]\) (IV) \(\lim _{n \rightarrow \infty} \frac{n^{k} \sin ^{2}(n !)}{n+2}, 0

Instructions: In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True \begin{aligned} &\text { Assertion: If } t_{r}=\frac{1^{2}+2^{2}+3^{2}+\ldots+r^{2}}{1^{3}+2^{3}+3^{3}+\ldots+r^{3}} \text { and }\\\ &S_{n}=\sum_{r=1}^{n}(-1)^{r} \cdot t_{r}, \text { then } \lim _{n \rightarrow \infty} S_{n}=\frac{2}{3}\\\ &\text { Reason: } 1^{2}+2^{2}+3^{2}+\ldots+r^{2}=\frac{r(r+1)(2 r+1)}{6}\\\ &\text { and } 1^{3}+2^{3}+3^{3}+\ldots+r^{3}=\left(\frac{r(r+1)}{2}\right)^{2} \end{aligned}

Instructions: In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If \(x_{1}=3\) and \(x_{n+1}=\sqrt{2+x_{n}}, n \geq 1\), then \(\lim _{n \rightarrow \infty} x_{n}=2\) Reason: A monotonically decreasing sequence which is bounded below is convergent

Instructions: In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{(n-1) / n}\right)=e-1\) Reason: \(1+r+r^{2}+\ldots+r^{n-1}\) \(=\left\\{\begin{array}{l}\frac{1-r^{n}}{1-r} \text { if } r<1 \\\ \frac{r^{n}-1}{r-1} \text { if } r>1\end{array}\right.\)

Instructions: In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\lim _{x \rightarrow 0} \frac{e-(1+x)^{1 / x}}{x}=\frac{e}{2}\) Reason: \(\lim _{x \rightarrow 0} \frac{\ln (1+x)-x}{x}=0\) and \(\lim _{x \rightarrow 0} \frac{\ln (1+x)-x}{x^{2}}=-\frac{1}{2}\)

Instructions: In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\lim _{n \rightarrow \infty} \frac{1}{2} \tan \frac{x}{2}+\frac{1}{2^{2}} \tan \frac{x}{2^{2}}+\ldots+\frac{1}{2^{n}} \tan \frac{x}{2^{n}}\) \(=-\cot x+\frac{1}{x}\) Reason: \(\cot x+\frac{1}{2} \tan \frac{x}{2}=\frac{1}{2} \cot \frac{x}{2}\)

Instructions: In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(\lim _{\theta \rightarrow 0} \frac{\cot \theta \tan ^{-1}(m \tan \theta)-m \cos ^{2}(\theta / 2)}{\sin ^{2}(\theta / 2)}\) \(=m-\frac{4}{3} m^{3}\) Reason: \(\lim _{x \rightarrow 0} \frac{x-\tan x}{x^{3}}=\frac{1}{3}\)

\(\lim _{x \rightarrow \infty} \frac{\sqrt{1-\cos 2 x}}{\sqrt{2} x}\) is (A) 1 (B) \(-1\) (C) Zero (D) Does not exist

\(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+5 x+3}{x^{2}+x+2}\right)^{x}\) is equal to (A) \(e^{4}\) (B) \(e^{2}\) (C) \(e^{3}\) (D) \(e\)

For \(x \in \mathrm{R}, \lim _{x \rightarrow \infty}\left(\frac{x-3}{x+2}\right)^{2}\) is equal to (A) \(e\) (B) \(e^{-1}\) (B) \(e^{-5}\) (D) \(e^{5}\)

Let \(f(2)=4\) and \(f^{\prime}(2)=4\). Then \(\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}\) is given by (A) 2 (B) \(-2\) (C) \(-4\) (D) 3

\(\lim _{x \rightarrow \pi / 2} \frac{\left[1-\tan \left(\frac{x}{2}\right)\right][1-\sin x]}{\left[1+\tan \left(\frac{x}{2}\right)\right][\pi-2 x]^{3}}\) is (A) \(\frac{1}{8}\) (B) 0 (C) \(\frac{1}{32}\) (D) \(\infty\)

If \(\lim _{x \rightarrow 0} \frac{\log (3+x)-\log (3-x)}{x}=k\), the value of \(k\) is (A) 0 (B) \(-\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) \(-\frac{2}{3}\)

Let \(f(a)=g(a)=k\) and their \(n^{\text {th }}\) derivatives \(f^{n}(a)\), \(g^{n}(a)\) exist and are not equal for some \(n .\) Further if \(\lim _{x \rightarrow a} \frac{f(a) g(x)-f(a)-g(a) f(x)+g(a)}{g(x)-f(x)}=4\), then the value of \(k\) is (A) 4 (B) 2 (C) 1 (D) 0

If \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}+\frac{b}{x^{2}}\right)^{2 x}=e^{2}\), then the values of \(a\) and \(b\), are (A) \(a \in \underline{\underline{R}}, b \in \underline{\underline{R}}\) (B) \(a=1, b \in \underline{R}\) (C) \(a \in \underline{R}, b=\underline{R}\) (D) \(a=1\) and \(b=2\)

Let \(\alpha\) and \(\beta\) be the distinct roots of \(a x^{2}+b x+c=0\), then \(\lim _{x \rightarrow \alpha} \frac{1-\cos \left(a x^{2}+b x+c\right)}{(x-\alpha)^{2}}\) is equal to (A) \(\frac{a^{2}}{2}(\alpha-\beta)^{2}\) (B) 0 (C) \(-\frac{a^{2}}{2}(\alpha-\beta)^{2}\) (D) \(\frac{1}{2}(\alpha-\beta)^{2}\)

Let \(f: R \rightarrow R\) be a positive increasing function such that \(\lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 .\) Then, \(\lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=\) (A) \(\frac{2}{3}\) (B) \(\frac{3}{2}\) (C) 3 (D) 1

Limit of \(\left(\frac{\sqrt{1-\cos \\{2(x-2)\\}}}{x-2}\right)\) as \(x\) tends to 2 (A) equals \(\sqrt{2}\) (B) equals \(-\sqrt{2}\) (C) equals \(\frac{1}{\sqrt{2}}\) (D) does not exist

The value of \(\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x}\) is equal to (A) \(\frac{1}{2}\) (B) 1 (C) 2 (D) \(-\frac{1}{4}\)

Let \(f(x)\) be a forth degree polynomial having extreme values at \(x=1\) and \(x=2\). If \(\lim _{x \rightarrow 0}\left[1+\frac{f(x)}{x^{2}}\right]=3\), then \(f(2)\) is equal to (A) \(-4\) (B) 0 (C) 4 (D) \(-8\)

The value of \(\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x}\) is equal to (A) 3 (B) 2 (C) \(\frac{1}{2}\) (D) 4

Let \(p=\lim _{x \rightarrow 0+}\left(1+\tan ^{2} \sqrt{x}\right)^{\frac{1}{2 x}}\) then \(\log p\) is equal to (A) \(\frac{1}{4}\) (B) 2 (C) 1 (D) \(\frac{1}{2}\)