Question

If $\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=10$ and $\underset{x\to a}{lim}f\left(x\right)=2$, then find the value of $\underset{x\to a}{lim}g\left(x\right)$, provided that $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exists ___

Answer 1

If the limits $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exist, then $\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$

$\Rightarrow 10=2+\underset{x\to a}{lim}g\left(x\right)$

$\Rightarrow \underset{x\to a}{lim}g\left(x\right)=8$