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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.

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1 Answer

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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 $x$

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