Answer
False
Work Step by Step
Let \[f(x)=1+\frac{1}{x}\]
\[\Rightarrow \lim_{x\rightarrow \infty}f(x)=1\]
and \[g(x)=x\]
\[\Rightarrow \lim_{x\rightarrow \infty}g(x)=\infty\]
Consider \[\left[f(x)\right]^{g(x)}=\left(1+\frac{1}{x}\right)^x\]
\[\Rightarrow \lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x=e^{\lim_{x\rightarrow \infty}x\left[1+\frac{1}{x}-1\right]}\]
\[\Rightarrow \lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x=e\neq 1\]
