TITLE: any number raised to the power of infinity
QUESTION [5 upvotes]: 1) I saw in a book that "the limit as $x$ approaches positive infinity of $e^x$ equals $0$" I want to ask about this?
2) if the $a$ is a negative number and we take a limit like "the limit as $x$ approaches positive infinity of $a^x$ equals?" and if $x$ approaches minus infinity then what happens?
 Please also tell me what would happen if $a$ is positive number.

REPLY [6 votes]: As long as the base is greater than one, the same thing happens.  $$\lim_{x \to \infty}a^x=\infty, \lim_{x \to -\infty}a^x=0$$ for any $a \gt 1$.  $$\lim_{x \to +\infty}a^x=0, \lim_{x \to -\infty}a^x=\infty$$ for any $0 \lt a \lt 1$.  
For $a \lt 0$,  $a^x$ is undefined in the reals for irrational $x$