G Tewani Solutions for Chapter: Continuity and Differentiability, Exercise 2: DPP

If $g\left(x\right)=\underset{m\to \infty }{\lim }\frac{x^{m}f\left(x\right)+h\left(x\right)+3}{2x^m+4x+1}$ when $x\ne 1$ and $g\left(1\right)=e^3$ such that $f\left(x\right), g\left(x\right)$ and $h\left(x\right)$ are continuous functions at $x=1$, then the value of $5f\left(1\right)-2h\left(1\right)$ is:

The number of points of discontinuity of $f\left(x\right)={\left[2x\right]}^2-{\left\{2x\right\}}^2$ (where, $\left[·\right]$ denotes the greatest integer function and $\left\{·\right\}$ is fractional part) of $x$ in the interval $\left(-2,2\right)$, are

Let $f\left(x\right)=\left\{\begin{array}{cc}-3+\left|x\right|,& -\infty <x<1\\ a+|2-x|,& 1\le x<\infty \end{array}\right.$ and $g\left(x\right)=\left\{\begin{array}{cc}2-|-x|,& -\infty <x<2\\ -b+sgn\left(x\right),& 2\le x<\infty \end{array}\right.$ where, $sgn\left(x\right)$ denotes signum function of $x.$

If $h\left(x\right)=f\left(x\right)+g\left(x\right)$ is discontinuous at exactly one point, then which of the following is not possible?

The function $f\left(x\right)=\frac{x^3}{8}-\sin \pi x+4$ in $\left[-4, 4\right]$ does not take the value

Let $f\left(x\right)$ be the continuous function $f: R\to R$ satisfying $f\left(0\right)=1$ and $f\left(2x\right)-f\left(x\right)=x$. Then the value of $f\left(3\right)$ is:

Given $f\left(x\right)=\left\{\begin{array}{ll}3-\left[{\cot }^{-1}\left(\frac{2x^3-3}{x^2}\right)\right]& \mathrm{for} x>0\\ \left\{x^2\right\}\cos \left(e^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)& \mathrm{for} x<0\end{array}\right.$, (where, $\{·\}$ and $[·]$ denotes the fractional part and the integral part functions respectively).

Then which of the following statements do/does not hold good?

Let $f\left(x\right)=\left\{\begin{array}{ll}x\left[\frac{1}{x}\right]+x\left[x\right]& \mathrm{if} x\ne 0\\ 0& \mathrm{if} x=0\end{array}\right.$, (where, $\left[x\right]$ denotes the greatest integer function). Then, the correct statements is/are

A function $f:R\to R$ is defined as

 $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{a{x}^2+bx+c+e^{nx}}{1+c·e^{nx}}$, where $f$ is continuous on $R$, then