T-EB ) Find lim f(2). Do not use quep ל -ולvלד T-人ス det fézf 32+7

Given function is $f\left(x\right)=\left\{\begin{array}{l}3x+7ifx<-1\\ x^2+3if-1\le x<1\\ 5x-1if1\le x\end{array}\right.$.

(a) We have to find the $\underset{x\to -1}{\lim }f\left(x\right)$.

Left hand limit at $x=-1$ is given by

$\begin{array}{c}\underset{x\to -1^{-}}{\lim }f\left(x\right)=\underset{x\to -1^{-}}{\lim }\left(3x+7\right)\\ =4\end{array}$

Hence, $\underset{x\to -1^{-}}{\lim }f\left(x\right)=4$

Right hand limit at $x=-1$ is given by

$\begin{array}{c}\underset{x\to -1^{+}}{\lim }f\left(x\right)=\underset{x\to -1^{+}}{\lim }\left(x^2+3\right)\\ =4\end{array}$

Hence, $\underset{x\to -1^{+}}{\lim }f\left(x\right)=4$

Hence at $x=-1$, limit of $f\left(x\right)$ is exist

And $\underset{x\to -1}{\lim }f\left(x\right)=4$