Why is $\lim\limits_{x\to a} e^x = e^{\lim\limits_ {x\to a} x}$?

Question

This is a confusion that I have had for a long time. " Why is $\lim\limits_{x\to a} e^x = e^{\lim\limits_ {x\to a} x}$?"

Is there any proof or logic behind? Please explain. I have googled this and I have not received any satisfying answers

If you want to know why $e^x$ is continuous, you'll need to provide a definition of the function $e^x$ that you're comfortable with. For example, is it $\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!}$? Or maybe it's the unique solution to the IVP $y' = y, y(0) = 1$? — Theo Bendit, May 15 '19 at 06:50
I do not think this question should be closed. In particular, I don't think it is "missing context of other details". Its just a thought someone is having, and its the kind of thought which admits a good answer in the form of a reference to a book or some course notes. — user1729, May 15 '19 at 11:18
Here is a relevant post on the definition of $a^x$: What does $2^x$ really mean when $x$ is not an integer?. For the result mentioned by Siong Thye Goh, see Proving that $\lim\limits_{ x \rightarrow 0}{f(g(x))}=f(\lim\limits_{ x \rightarrow 0}g(x))$ when $f$ is continuous. — YuiTo Cheng, May 15 '19 at 15:48
@TheoBendit: It's definitely not the quadratic you provided! =P — user21820, May 16 '19 at 10:10
Until you define precisely what "$e^x$" means, your question cannot really be answered properly. Separate from that, I strongly recommend that you start learning proper real analysis from a proper textbook such as Spivak's Calculus. — user21820, May 16 '19 at 10:18

score 7 · Accepted Answer · answered May 15 '19 at 06:41

7

For continuous function, the function of the limit is the limit of the function.

$$f(\lim_{x \to a}x) = \lim_{x \to a}f(x)$$

Exponentiation is a continuous function.

answered May 15 '19 at 06:41

Siong Thye Goh

Please be more precise. Exponentiation is continuous on a certain domain, not everywhere that it is useful to be defined. – user21820 May 16 '19 at 10:20

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