A Question About the possible values of 1^infinity

Question

if we have

$$\lim_{x\to c}f(x) = 1 $$

$$\lim_{x\to c}g(x) = \infty $$

then can $\lim_{x\to c}f(x)^{g(x)} = -\infty $ ever be true?

If so, what are some examples? If not, would it be different if $ \lim_{x\to c}g(x) = -\infty $ ?

Answer 1

Under your hypotheses, $\lim_{x \to c} f(x)^{g(x)}$ can not be $-\infty$. To see this, for $x$ sufficiently close to $c$, $f(x)$ is positive so that $f(x)^{g(x)}$ is non-negative for all $x$ in a neighborhood of $c$ and so cannot have limit $-\infty$ as $x \to c$.

Note this works for both $\lim_{x \to c} g(x) = \infty$ and $-\infty$.