Why is it justified to move the limit into the exponent?

Question

On my last math test my teacher told me that my notation for evaluating limits "might be problematic." The notation he is referring to is when evaluating a limit of the form $$\lim_{x\to \infty}{f(x)}^{g(x)}$$ What I usually do is write it as $$\exp(\,\lim_{x\to \infty}g(x)\ln f(x)\,)$$ He would like me to write $$y=\lim_{x\to \infty}{f(x)}^{g(x)} \\ \ln y=\lim_{x\to \infty}{g(x)}\ln{f(x)}$$ They are essentially the same process, so why would my method be problematic? If there is nothing wrong with it, then how can I justify it to him? Specifically, how can I rigorously justify transferring the limit to the exponent? i.e. $$\lim_{x \to \infty} e^{f(x)}=\exp (\lim_{x \to \infty} f(x))$$

Answer 1

The property you need for $\lim_{x\to c}f(g(x))=f(\lim_{x\to c}g(x))$ to hold is for $f(x)$ to be continuous. This is either the definition of $f$ being a continuous function, or equivalent to the definition (some people use that the inverse image maps open sets to open sets). Since $e^x$ is continuous, you're fine.