Subject Experts Solutions for Chapter: Limits, Continuity and Differentiability, Exercise 1: Practice Exercise

Consider the following statements

I. $f\left(x\right)=\left[x\right],$ where $\left[.\right]$ is the greatest integer function, is discontinuous at $x=n$, where $n\in Z$.

II. $f\left(x\right)=\cot x$ is discontinuous at $x=n\pi ,$ where $n\in Z$.

Which of the above statement(s) is / are correct?

If $f\left(x\right)=\sqrt{25-x^2},$ then what is $\underset{x\to 1}{\lim }\frac{f\left(x\right)-f\left(1\right)}{x-1}$ equal to?

If $f\left(x\right)=\frac{\sin \left(e^{x-2}-1\right)}{\ln \left(x-1\right)}$, then $\underset{x\to 2}{\lim }f\left(x\right)$ is equal to

The function $f\left(x\right)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$ is not defined at $x=\pi$. The value of $f\left(\pi \right),$ so that $f\left(x\right)$ is continuous at $x=\pi ,$ is

Consider the following functions
I. $f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{x},& \text{ if }x\ne 0\\ 0,& \text{ if }x=0\end{array}\right.$
II. $f\left(x\right)=\left\{\begin{array}{ll}2x+5,& \text{ if }x>0\\ x^2+2x+5,& \text{ if }x\le 0\end{array}\right.$

Which of the above function(s) is/are derivable at $x=0 ?$

Consider the function $f\left(x\right)=\left\{\begin{array}{ccc}ax-2,& \text{ for }& -2<x<-1\\ -1,& \text{ for }& -1\le x\le 1\\ a+2\left(x-1\right){}^2,& \text{ for }& 1<x<2\end{array}\right.$. What is the value of $a$ for which $f\left(x\right)$ is continuous at $x=-1$ and $x=1 ?$

If $\underset{x\to 0}{\lim }\varphi \left(x\right)=a^2,$ where $a\ne 0,$ then what is $\underset{x\to 0}{\lim }\varphi \left(\frac{x}{a}\right)$ equal to?

What is $\underset{x\to 0}{\lim }{e}^{-\frac{1}{x^2}}$ equal to?