Is the fuction $f(x) = \frac{7}{e^x - 2}$ continuous on the interval $ [-1,1]$

Question

Hi I am having trouble trying to figure this problem out. I have tried to separate the equation into a piece-wise function but was no help in making things clear. I am not sure if I use the limit as $x$ approaches $0$ since its in the interval given, or do something else. Thanks.

Answer 1

Hint: what happens when $x\rightarrow\ln 2$?

Answer 2

Hint: Check if the denominator becomes equal to $0$ in the given interval: $$\begin{align*}e^x-2=0 &\iff e^x=2 \qquad \quad\mid \ln \\ &\iff \ln(e^x)=\ln2 \\&\iff x=\ln2\end{align*}$$ since $\ln$ and $e$ cancel out. Now, check whether $\ln 2 \in (-1, 1)$ or not and if yes, proceed as you know (with the limit).