Chapter 3

Given \(\begin{aligned} f(x) &=(1+x)^{\frac{1}{x}}, \quad x \neq 0 \\ &=e, \quad x=0 \end{aligned}\)

Given \(\begin{aligned} f(x) &=x \sin \left(\frac{1}{x}\right) \quad x \neq 0 \\\ &=0, \quad x=0 \end{aligned}\)

Given \(f(x)=\frac{e^{x}-1}{x}, \quad x \neq 0\) \(=e, \quad x=0 .\)

Given \(\begin{aligned} f(x) &=\frac{\sin 3 x}{x}, \quad x \neq 0 \\ &=1, \quad x=0 \end{aligned}\)

Given \(\begin{aligned} f(x) &=\frac{1}{5}\left(2 x^{2}+3\right), \quad x \leq 1 \\\ &=6-5 x, \quad 1

Given \(\begin{aligned} f(x) &=\frac{\cos x-\sin x}{\cos 2 x}, \quad x \neq \frac{\pi}{4} \\ &=\frac{1}{\sqrt{2}}, \quad x=\frac{\pi}{4} . \end{aligned}\)

Check the function \(\begin{aligned} f(x) &=\frac{\cos x}{\frac{\pi}{2}-x}, \quad x \neq \frac{\pi}{2} \\ &=1, \quad x=\frac{\pi}{2} \end{aligned}\)

Given \(\begin{aligned} f(x) &=\frac{\sqrt{1+x}-\sqrt[3]{1+x}}{x}, \quad x \neq 0 \\ &=\frac{1}{6}, \quad x=0 . \end{aligned}\)

Given \(\begin{aligned} f(x) &=0, \quad x<0 \\ &=x, \quad 0 \leq x<1 \\\ &=-x^{2}+4 x-2, \quad 1 \leq x<3 \\ &=4-x, \quad x \geq 3 \end{aligned}\)

A function \(f(x)\) is defined as \(f(x)=\frac{x^{2}-4 x+3}{x^{2}-1}, x \neq 1\) \(=2, \quad x=1 .\)

Construct the graph of the function given below $$ \begin{aligned} f(x) &=x-1, & x<0 \\ &=\frac{1}{4}, & x &=0 \\ &=x^{2}, & x>0 . \end{aligned} $$

Let \(\begin{aligned} f(x) &=\frac{x^{2}-1}{x^{2}-2|x-1|-1}, \quad x \neq 1 \\\ &=\frac{1}{2}, \quad x=1 \end{aligned}\)

For what value of \(a\) will the function \(f(x)\) be continuous at \(x=1\) $$ \begin{aligned} f(x) &=x+1, \quad x \leq 1 \\ &=3-a x^{2}, \quad x>1 .\\{\text { Ans } a=1\\} \end{aligned} $$

Let \(\begin{aligned} f(x) &=\frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, \quad x \neq 2 \\ &=k, \quad x=2 . \end{aligned}\)

\(\begin{aligned} f(x) &=\frac{1-\cos 4 x}{x^{2}}, x<0 \\ &=a, \quad x=0 \\\ &=\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, \quad x>0 . \end{aligned}\)

For what value of \(a\), the function \(\begin{aligned} f(x) &=x^{a} \sin \frac{1}{x}, \quad x \neq 0 \\ &=0, \quad x=0 \end{aligned}\)

Choose \(A\) and \(B\) so as to make the function \(f(x)\) continuous at \(x=\pm \frac{\pi}{2}\) $$ \begin{aligned} f(x) &=-2 \sin x, \quad x \leq-\frac{\pi}{2} \\ &=A \sin x+B, \quad-\frac{\pi}{2}

Let \(\begin{aligned} f(x) &=(1+|\sin x|) \sqrt{\sin x}, \quad-\frac{\pi}{6}

Determine \(a, b\) and \(c\) for which the function \(f(x)=\frac{\sin (a+1) x+\sin x}{x}, x<0\) \(=c, \quad x=0\) \(=\frac{\left(x+b x^{2}\right)^{\frac{1}{2}}-x^{\frac{1}{2}}}{b x^{\frac{2}{2}}}, \quad x>0\)

Given the function \(\begin{aligned} f(x) &=\frac{a^{\sin x}-a^{\operatorname{lan} x}}{\tan x-\sin x}, x>0 \\ &=\frac{\ln \left(1+x+x^{2}\right)+\ln \left(1-x+x^{2}\right)}{\sec x-\cos x}, x<0 . \end{aligned}\) If \(f(x)\) is continuous at \(x=0\), find the value of \(a\). Now, \(g(x)=\ln \left(2-\frac{x}{a}\right) \cdot \cot (x-a), x \neq a\). If \(g(x)\) is

Given \(\begin{aligned} f(x) &=1, \quad x=0 \\ &=x, \quad 0

Given \(\begin{aligned} f(x) &=\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}}, \quad x<0 \\\ &=\frac{1}{\sqrt{e}}, \quad x=0 \\ &=(1+\ln (\cos (\sin x)))^{\frac{1}{x}}, \quad x>0 . \end{aligned}\)

Find the constant \(A\) such that the function \(\begin{aligned} f(x) &=\frac{e^{4 x}-1}{\tan x}, \quad x<0 \\ &=A^{2}-7 A+16, \quad x=0 \\ &=A, \quad x>0 \end{aligned}\)

Define the following functions at \(x=0\) so as to make them continuous:- i. \(f(x)=\frac{5 x^{2}-3 x}{2 x}\) ii. \(f(x)=\frac{\sqrt{1-\cos x}}{x}\) iii. \(f(x)=\frac{\ln (1+x)-\ln (1-x)}{x}\) iv. \(f(x)=\frac{2-\sqrt{x+4}}{\sin 2 x}\) v. \(f(x)=\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}\\{\) vi. \(f(x)=\frac{1-\cos x}{\sin ^{2} x}\) vii. \(f(x)=(x+1)^{\text {cotr }}\). \(\\{\)

Prove the continuity of the following functions by first principles:- i. \(f(x)=x^{n}\) ii. \(\quad f(x)=\frac{1}{x}\) iii. \(f(x)=e^{x}\) iv. \(f(x)=\ln x\) v. \(f(x)=\sin x\) vi. \(f(x)=x \sin x\) vii. \(f(x)=\cos x \cdot \ln x\)

If \(f(x+y)=f(x)+f(y) \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then show that \(f(x)\) is continuous \(\forall x\).

If \(f(x+y)=f(x) \cdot f(y) \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then show that \(f(x)\) is continuous \(\forall x\).

If \(f(x y)=f(x)+f(y) \forall x, y \neq 0\) and \(f(x)\) is continuous at \(x=1\), then check the continuity of \(f(x)\).

If \(f(x y)=f(x) \cdot f(y) \forall x, y\) and \(f(x)\) is continuous at \(x=1\), then show that \(f(x)\) is continuous for all \(x\) except \(x=0\).

If \(f(x+y)=f(x)+f(y) \forall x, y\) and \(f(x)\) is continuous at a point \(x=a\), then show that \(f(x)\) is continuous \(\forall x .\)

If \(f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).

If \(f(x+2 y)=f(x)+2 f(y)-2 f(0) \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).

If \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)+f(0)}{3} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).

If \(f\left(\frac{x+2 y}{3}\right)=\frac{f(x)+2 f(y)}{3} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).

Show that a polynomial function \(P_{n}(x)=a_{n} x^{n}+a_{s-1} x^{n-1}+\ldots \ldots+a_{1} x+a_{0}\) is a continuous function.

Show that a rational function is a continuous function.

Test the following functions for continuity \(f(x)=\frac{\cos (\ln x)+\sin ^{3} x \cdot \tan ^{-1} x}{e^{x}+\cosh x}\).

Check the continuity of \(|x|\) and \(\operatorname{sgn} x\).

Test the following functions for continuity $$ \begin{aligned} \phi(x) &=0, \quad x=0 \\ &=\frac{1}{2}-x, \quad 0

Find the values of \(a\) and \(b\) so that the function \(\begin{aligned} f(x) &=x+a \sqrt{2} \sin x, \quad 0 \leq x<\frac{\pi}{4} \\ &=2 x \cot x+b, \quad \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ &=a \cos 2 x-b \sin x, \quad \frac{\pi}{2}

Given the function \(f(x)=\frac{1}{1-x}\). Find the points of discontinuity of the function \(f(x), f(f(x))\) \& \(f(f(f(x)))\). \(\\{\)

Let \(f(x)=\frac{1}{x-1}\) \& \(g(x)=\frac{1}{x^{2}+x-2}\). Find the points where \(g(f(x))\) is discontinuous.

Given \(f(x)=x-1, \quad x \geq 0\) \(=x+1, \quad x<0 .\) Test the function \(\phi(x)=[f(x)]^{2}\) for continuity.

Investigate the functions \(f(g(x))\) and \(g(f(x))\) for continuity if \(f(x)=\operatorname{sgn} x\) and \(g(x)=x\left(1-x^{2}\right)\).

Given \(f(x)=1+x, \quad 0 \leq x \leq 2\) \(=3-x, \quad 2

Show that \(f(x)=1, \quad x\) is rational \(=-1, \quad x\) is irrational is discontinuous for all \(x\).

Given \(f(x)=x, \quad x\) is rational \(=-x, \quad x\) is irrational Show that \(f(x)\) is continuous at \(x=0\) only.

Test the function for continuity $$ \begin{aligned} f(x) &=\frac{1}{2^{n}}, \frac{1}{2^{n+1}}

Discuss the continuity of the function \(f(x)=\lim _{n \rightarrow \infty} \frac{(1+\sin \pi x)^{n}-1}{(1+\sin \pi x)^{n}+1}\) at the point \(x=1\).

Check the continuity of the function \(f(x)=\lim _{n \rightarrow \infty} \frac{\log (2+x)-x^{2 n} \sin x}{1+x^{2 \pi}}\).