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

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

• It's what we call continuity. – Lord Shark the Unknown May 15 at 6:40
• 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 at 6:50
• 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 at 11:18
• @TheoBendit: It's definitely not the quadratic you provided! =P – user21820 May 16 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 at 10:18

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