Question

If $f\left(x\right)=\{\begin{array}{c}\cos x;x\ge 0\\ x+k;x<0\end{array}$ find the value of $k$ if $f\left(x\right)$ is continuous at $x=0$

Answer 1

$f\left(x\right)=\{\begin{array}{c}\cos x;x\ge 0\\ x+k;x<0\end{array}$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}cosx=1$

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

So for $f\left(x\right)$ to be continuous $k=1$