Question

If $f\left(x\right)\{\begin{array}{c}m{x}^2+n,x<0\\ nx+m,0\le x\le \\ n{x}^3+m,x>1\end{array}1$,

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.

Answer 1

For a limit to exist there should exist both left hand limit and right hand limit and both limits must be equal

For $\underset{x\to 0}{lim}f\left(x\right)$ to exist, the condition is;

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}f\left(x\right)=f\left(0\right)$

Applying above conditions for $\underset{x\to 0}{lim}$, we get

$m\left(0\right)+n=n\left(0\right)+m$

$\Rightarrow m=n$ ...... $\left(1\right)$

Applying similar condition for $\underset{x\to 1}{lim}$ we get ;

$n\left(1\right)+m=n\left(1^3\right)+m$

$\Rightarrow n+m=n+m$ which is always true

For both the limits to exist, $m$ must be equal to $n$

$\therefore m=n$