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Math Help - Limit of e^x

  1. #1
    Senior Member Paze's Avatar
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    Limit of e^x

    When working with a limit of e^x such as:

    The limit as x approaches infinity of \frac{x^2}{1-e^x}

    Can I simply state that it tends towards 1/((e^inf)/inf) and therefore to 0, or is the limit not yet defined since it tends towards inf/inf below 1? Thanks.
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    Re: Limit of e^x

    Why do you think that 1/((e^inf)/inf) is 0? In general, you can't compare infinities.
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    Re: Limit of e^x

    Quote Originally Posted by emakarov View Post
    Why do you think that 1/((e^inf)/inf) is 0? In general, you can't compare infinities.
    So I'll need l'hopital right?
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    Re: Limit of e^x

    Yes, you could use L'Hopital's rule. To find limits, you need some database of standard facts, e.g., \frac{a_nx^n+\dots+a_0}{b_nx^n+\dots+b_0}\to\frac{  a_n}{b_n} as x\to\infty provided a_n\ne0 and b_n\ne0. The fact that the \lim_{x\to\infty}\frac{x^n}{a^x}=0 where a>1 is one of those standard facts, but one has to prove it for the first time.
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    Re: Limit of e^x

    You could also use the MacLaurin series for e^x,  1+ x+ \frac{x^2}{2}+ \frac{x^3}{6}+ \cdot\cdot\cdot so that 1- e^x= -(x+ \frac{x^2}{2}+ \frac{x^3}{6}+ \cdot\cdot\cdot}.

    Now you have \frac{x^2}{1- e^x}= -\frac{x^2}{x+ \frac{x^2}{2}+ \frac{x^3}{6}+ \cdot\cdot\cdot}.

    Divide both numerator and denominator by x^2 to get -\frac{1}{\frac{1}{x}+ \frac{1}{2}+ \frac{x}{6}+ \cdot\cdot\cdot} which goes to infinity as x goes to infinity. Notice that, by this argument, \lim_{x\to\infty} \frac{x}{1- e^x}= 1.
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    Re: Limit of e^x

    Quote Originally Posted by HallsofIvy View Post
    You could also use the MacLaurin series for e^x,  1+ x+ \frac{x^2}{2}+ \frac{x^3}{6}+ \cdot\cdot\cdot so that 1- e^x= -(x+ \frac{x^2}{2}+ \frac{x^3}{6}+ \cdot\cdot\cdot}.

    Now you have \frac{x^2}{1- e^x}= -\frac{x^2}{x+ \frac{x^2}{2}+ \frac{x^3}{6}+ \cdot\cdot\cdot}.

    Divide both numerator and denominator by x^2 to get -\frac{1}{\frac{1}{x}+ \frac{1}{2}+ \frac{x}{6}+ \cdot\cdot\cdot} which goes to infinity as x goes to infinity. Notice that, by this argument, \lim_{x\to\infty} \frac{x}{1- e^x}= 1.
    According to calculators and l'hopitals, we have it tending towards 0. Which one is correct...?
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    Re: Limit of e^x

    Quote Originally Posted by Paze View Post
    According to calculators and l'hopitals, we have it tending towards 0. Which one is correct...?
    In both cases the limit is 0.
    I myself find Prof. Ivey's post confusing. But his point is valid: the series approach gives that limit.
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    Re: Limit of e^x

    Quote Originally Posted by Plato View Post
    In both cases the limit is 0.
    I myself find Prof. Ivey's post confusing. But his point is valid: the series approach gives that limit.
    Interesting. I'm going to expand on this idea for my assignment (I hate regurgitating trivial information from a blackboard and I don't care if I get downgraded for veering off course)
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    Re: Limit of e^x

    I think you cannot use the Maclaurin series for $e^x$ which is about $x=0$ to analyze $e^x$ when $x\to\infty$. The expansion point (zero) is too far away from the limit point (infinity).

    To solve the problem, you have to use L'Hospital's or just know that exponentials grow faster than polynomials.

    This statement above is false: $\displaystyle\lim_{x\to\infty}\frac{x}{1-e^x}=-1$. The Maclaurin series approach gives an erroneous answer, because it cannot be used here.

    Rather, the limit equals 0, as pointed out in other posts.
    Last edited by limiTS; May 5th 2015 at 08:33 PM.
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    Re: Limit of e^x

    Quote Originally Posted by emakarov View Post
    The fact that the \lim_{x\to\infty}\frac{x^n}{a^x}=0 where a>1 is one of those standard facts, but one has to prove it for the first time.
    $$\begin{aligned}
    0 &\lt {1 \over t} \lt t^{c-1} \qquad (t \gt 1,\, c \gt 0) \\
    \int_1^x 0 \,\mathrm d t &\lt \int_1^x {1 \over t} \,\mathrm d t \lt \int_1^x t^{c-1} \,\mathrm d t \qquad (x \gt 1) \\
    0 &\lt \log x \lt {x^c - 1 \over c} \lt {x^c \over c} \\
    0 &\lt \log^a x \lt {x^{ac} \over c^a} \qquad (a \gt 0) \\
    0 &\lt {\log^a x \over x^b} \lt {x^{ac-b} \over c^a} \qquad (b \gt 0) \\
    \end{aligned}$$
    Since this is true for all $c \gt 0$ we may pick any $c = {b \over 2a}$ so that
    $$
    0 \lt {\log^a x \over x^b} \lt {x^{-{b \over 2}} \over c^a} \\
    $$
    And now, taking the limit as $x \to \infty$ gives
    $$
    0 \le \lim_{x \to \infty} {\log^a x \over x^b} \le \lim_{x \to \infty} {x^{-{b \over 2}} \over c^a} = 0 \\ \implies \lim_{x \to \infty} {\log^a x \over x^b} = 0
    $$
    Then, writing $x = \mathrm e^y$ we have
    $$\begin{aligned}
    \lim_{x \to \infty} {\log^a x \over x^b} &= 0 \\
    \lim_{y \to \infty} {\log^a \mathrm e^y \over \mathrm e^{by}} &= 0 \\
    \lim_{y \to \infty} {y^a \over \mathrm e^{by}} &= 0 \\
    \end{aligned}$$
    Now, addressing your question
    $$\begin{aligned}
    \lim_{x \to \infty} {x^2 \over 1 - \mathrm e^x} &= \lim_{x \to \infty} {x^2 \mathrm e^{-x} \over \mathrm e^{-x} - 1} \\
    &= \lim_{x \to \infty} {{x^2 \over \mathrm e^{x}} \over \mathrm e^{-x} - 1} \\
    &= {0 \over 0 - 1} = 0
    \end{aligned}$$
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    Re: Limit of e^x

    Quote Originally Posted by Paze View Post
    When working with a limit of e^x such as:

    The limit as x approaches infinity of \frac{x^2}{1-e^x}

    Can I simply state that it tends towards 1/((e^inf)/inf) and therefore to 0, or is the limit not yet defined since it tends towards inf/inf below 1? Thanks.
    Is it x --> + infinity or - infinity. You could alternatively reason this way:

    as x --> + infinity (1 - e^x) --> (- e^x)
    At the limit the ratio varies as x^2/(-e^x) and the limit is 0 from the left

    as x --> - infinity (1 - e^x) --> 1
    At the limit the ratio varies as x^2 --> + infinity
    Last edited by votan; May 5th 2015 at 10:09 PM.
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    Re: Limit of e^x

    Quote Originally Posted by limiTS View Post
    I think you cannot use the Maclaurin series for $e^x$ which is about $x=0$ to analyze $e^x$ when $x\to\infty$.
    For a general function this is true, but the power series for $\mathrm e^x$ converges for all $x$, so in this case you can.
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