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

  • 11
    $\begingroup$ It's what we call continuity. $\endgroup$ – Lord Shark the Unknown May 15 at 6:40
  • 2
    $\begingroup$ 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 $y' = y, y(0) = 1$? $\endgroup$ – Theo Bendit May 15 at 6:50
  • 3
    $\begingroup$ 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. $\endgroup$ – user1729 May 15 at 11:18
  • 1
    $\begingroup$ @TheoBendit: It's definitely not the quadratic you provided! =P $\endgroup$ – user21820 May 16 at 10:10
  • 1
    $\begingroup$ 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. $\endgroup$ – user21820 May 16 at 10:18

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.