Question

Let $f\left(x\right)=\begin{array}{cc}x+2& x\le -1\\ c{x}^2& x>-1\end{array}$, find 'c' if $\underset{x\to -1}{lim}f\left(x\right)$ exists.

Answer 1

For $\underset{x\to -1}{lim}f\left(x\right)$ to exist,

$RHL(-1)=LHL(-1)=f(-1)$

$RHL$

$\Rightarrow \underset{x\to -1}{lim}c{x}^2$

$\Rightarrow \underset{h\to 0}{lim}c(x+h{)}^2$

$\Rightarrow \underset{h\to 0}{lim}c(-1+h{)}^2$

$\Rightarrow c$

Now,

$f\left(x\right)=x+2=1$

$f(-1)=(-1)+2=1$

Now,$f(-1)=RHL$

$\Rightarrow c=1$