Assam Board Solutions for Chapter: Limits and Derivatives, Exercise 1: EXERCISE 13.1

Evaluate:$\underset{ x\to 1}{\lim }\left(3-x^2\right)$

If $\underset{r\to 1}{\lim }\pi r^2=k\mathrm{\pi }$, then $\frac{k}{2}$.is

Evaluate:$\underset{x\to 0}{\lim }\frac{{\left(x+1\right)}^5-1}{x}$

Suppose $f\left(x\right)=\left\{\begin{array}{ll}a+bx,& x<1\\ 4,& x=1\\ b-ax,& x>1\end{array}\right.$ and if $\underset{x\to 1}{\lim }f\left(x\right)=f\left(1\right)$ what are possible values of $a$ and $b$?

Let $a_1, a_2, \dots , a_n$ be fixed real numbers and define a function 
$f\left(x\right)=\left(x-a_1\right)\left(x-a_2\right)\dots \left(x-a_n\right)$

What is $\underset{x\to a_1}{\lim }f\left(x\right)$? For some $a\ne a_1, a_2, \dots , a_n$, compute $\underset{x\to a}{\lim }f\left(x\right)$.

If $f\left(x\right)=\left\{\begin{array}{ll}\left|x\right|+1,& x<0\\ 0,& x=0\\ \left|x\right|-1,& x>0\end{array}\right.$.

For what value (s) of $a$ does $\underset{x\to a}{\lim }f\left(x\right)$ exists?

If the function $f\left(x\right)$ satisfies $\underset{x\to 1}{\lim }\frac{f\left(x\right)-2}{x^2-1}=\pi$, evaluate $\underset{x\to 1}{\lim }f\left(x\right)$.

If $f\left(x\right)=\left\{\begin{array}{lc}m{x}^2+n,& x<0\\ nx+m,& 0\le x\le 1\\ n{x}^3+m,& x>1\end{array}\right.$. For what integers $m$ and $n$ does both $\underset{x\to 0}{\lim }f\left(x\right)$ and $\underset{x\to 1}{\lim }f\left(x\right)$ exist?