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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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    There's no "a" in the quote you say you saw.2011-10-05
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    The limit of $e^x$ as $x$ approaches infinity is $\infty$. Did you mean $\lim_{x\to -\infty}e^x=0$ or $\lim_{x\to \infty}e^{-x}=0$? Also, if $a<0$ the limit $\lim_{x\to\infty}a^x$ is not defined since $a^{\frac{b}{2}}$ does not exist for any $b>0$. In other words even roots of $a$ do not exist.2011-10-05
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    i mean limit as x--->- infinity of e to the power x2011-10-05
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    and i also asked about limit as x--> + infinity of e to the power x2011-10-05
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    then my question was if we take any number in place of e and do the the same then what happens, if the number is positive and what happens and if the number is negative2011-10-05

1 Answers 1

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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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    its confusing, look you said if a>1 then the result will be zero, and you did not told about if a is beteen 0 and 1.2011-10-05
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    i am assuming from your answer that if a>1 then limit x--> + infinity or - infinity of a to the power x is = 02011-10-05
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    @Zia: No. If $a>1$ then $\lim_{x\to\infty}a^x=\infty$ and $\lim_{x\to -\infty}a^x=0$. If $0$\lim_{x\to\infty}a^x=0$ and $\lim_{x\to -\infty}a^x=\infty$, since $a^{-x}=\frac{1}{a^x}$. – 2011-10-05
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    now thats the answer i was looking for thank you2011-10-05
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    @ZiaurRahman: I added in the $-\infty$ items2011-10-05
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    For $0$\lim_{x \to -\infty}a^x=\infty$ ? – 2012-07-14
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    @SaurabhHota: Thanks. Fixed.2012-07-14
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    A nice answer; I note that you don't include the case of a=1 exactly. :)2012-09-01