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How can we prove that $$\lim_{x\rightarrow 0}\cfrac{e^x-1-x}{x^2}=\cfrac{1}{2}$$ Without using L'hopital rule, and Taylor expansions?

Thanks

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    $\begingroup$ What is your definition of $e^x$? What can you use about this function? $\endgroup$ – ajotatxe Aug 30 '16 at 14:40
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    $\begingroup$ There is no pre-calculus definition of $e^{x}$ $\endgroup$ – preferred_anon Aug 30 '16 at 14:47
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    $\begingroup$ @DanielLittlewood I'd also say that there is no pre-calculus definition of limit... $\endgroup$ – ajotatxe Aug 30 '16 at 14:51
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    $\begingroup$ It can be worked out if you use $e^x = \lim_{n \to \infty} \left (1 + \frac xn \right)^n$ presuming the limit exists in the first place. $\endgroup$ – Umberto P. Aug 30 '16 at 14:55
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    $\begingroup$ @ajotatxe we can only use the definition stated in Umberto P. comment $\endgroup$ – An old man in the sea. Aug 30 '16 at 14:56
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Use the limit laws and the binomial theorem: you have $$\frac{e^x - (1+x)}{x^2} = \lim_{n \to \infty} \left( \frac{ (1+ \frac xn)^n - (1+x)}{x^2} \right) = \lim_{n \to \infty} \sum_{k=2}^n \binom nk \frac{x^{k-2}}{n^k} \\ = \frac 12 + x \left( \lim_{n \to \infty} \sum_{k=3}^n \binom nk \frac{x^{k-3}}{n^k} \right)$$ provided that the limit $ \displaystyle e^x = \lim_{n \to \infty} \left(1 + \frac xn \right)^n$ is assumed to exist.

As a by-product of this computation you get that $$\lim_{n \to \infty} \sum_{k=3}^n \binom nk \frac{x^{k-3}}{n^k}$$ exists too. With $x=1$ this implies $$\sup_n \sum_{k=3}^n \binom nk \frac{1}{n^k} < \infty$$ and consequently if $|x| \le 1$ then $$\sup_{n} \left| \sum_{k=3}^n \binom nk \frac{x^{k-3}}{n^k} \right| \le \sup_{n} \sum_{k=3}^n \binom nk \frac{|x^{k-3}|}{n^k} \le \sup_n \sum_{k=3}^n \binom nk \frac{1}{n^k} < \infty.$$

So, if $|x| \le 1$ then $$\left| \frac{e^x - (1+x)}{x^2} - \frac 12 \right| \le |x| \sup_n \sum_{k=3}^n \binom nk \frac{1}{n^k}.$$

Now let $x \to 0$.

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  • $\begingroup$ Nothing wrong with that but for the main result we do not need to know that the sequence $(\sum_{k=3}^nx^{k-3}n^{-k}\binom {n}{k})_{n\geq 3} $converges: For $n\geq k$ we have $0<n^{-k}\binom {n}{k}\leq 1.$ So for $n\geq 3$ and $|x|\leq 1/2$ we have $ |\sum_{k=3}^nn^{-k}x^{n-3}\binom {n}{k}|\leq$ $ \sum_{k=3}^n|x|^{k-3}\leq$ $ \sum_{k=3}^n2^{-(k-3)}<2.$ So for $|x|\leq 1/2$ we have $|(e^x-(1+x)/x^2-\frac {1}{2}|\leq 2|x|.$ $\endgroup$ – DanielWainfleet Aug 31 '16 at 7:20
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For every natural $n\ge 2$ we have $$\lim_{x\to 0}\frac{\left(1+\frac xn\right)^n-x-1}{x^2}=\lim_{x\to0}\frac{1+n\frac xn+\binom n2\frac{x^2}{n^2}+x^3P(x)-x-1}{x^2}=\frac{n-1}{2n}$$ where $P(x)$ is a polynomial.

This alone does not imply that your limit is $1/2$. We need to assume that the function $$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$$ is defined in some interval $I$ around $0$ (that is, that the limit exists for every $x\in I$) and continuous.

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  • $\begingroup$ I don't see how this proves the result. You've shown $\lim_{x\to 0}\frac{(1+x/n)^n - x - 1}{x^2} = \frac{1}{2}$ for each $n.$ But why is $$\lim_{x\to 0}\lim_{n\to \infty}\frac{(1+x/n)^n - x - 1}{x^2}= \lim_{n\to \infty}\lim_{x\to 0}\frac{(1+x/n)^n - x - 1}{x^2}?$$ $\endgroup$ – zhw. Sep 3 '16 at 19:46
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Are derivatives allowed? Assuming the existence of $$L = \lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \lim_{x \to 0} \frac{\left(\frac{e^x - 1}x\right) - 1}x$$ it is equal to the derivative of the (continuous) function $$f(x) = \begin{cases} \frac{e^x - 1}x & x \neq 0 \\ 1 & x = 0 \end{cases}$$ at $x = 0$. Thus, we have (formally) $$L = \lim_{t \to 0}\left.\frac{d}{dx}\left(\frac{e^x - 1}x\right)\right|_{x = t} = \lim_{t \to 0} \left(\frac{e^t}t - \frac{e^t - 1}{t^2}\right) = \lim_{t \to 0} \left(\frac{e^t - 1}t - \frac{e^t - 1 - t}{t^2}\right) = 1 - L$$

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  • $\begingroup$ Nice! ${{{{}}}}$ $\endgroup$ – user00000 Aug 30 '16 at 19:32
  • $\begingroup$ Hi, in you 1st equality on the last line, you're assuming that (e^x-1)/x is differentiable for any t in a neighbourhood of zero? and implicitly assuming that (e^x-1)/x is diff. also at zero? this is exactly what we want to prove... $\endgroup$ – An old man in the sea. Aug 31 '16 at 19:13
  • $\begingroup$ @oldman (that sounds derogatory, but I don't know what else to call you!), I believe I am assuming that $(e^x-1)/x$ is differentiable at zero, since that is equivalent to assuming that the limit exists. It is certainly differentiable in a deleted neighborhood of 0, as it is elementary and continuous off 0, but I did not explain why the derivative extends to 0, and why the derivative extends continuously to 0. Then again, such matters may be beyond the grasp of truly beginner students. $\endgroup$ – ALGebraist Sep 1 '16 at 3:29
  • $\begingroup$ ALGebraist, there's no problem, I'm an old man(or becoming one =D). When you try to tag someone with @ don't cut their usernames, otherwise they will not be notified. For e.g. you should have written @anoldmaninthesea (you can use the tab key to speed it up) $\endgroup$ – An old man in the sea. Sep 12 '16 at 10:38
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We have that $f(x)=e^x$ is an increasing convex function, hence for any $\varepsilon>0$ there is a neighbourhood of the origin $U$ where the inequality $$ (1-\varepsilon)x\leq e^x-1 \leq (1+\varepsilon)x \tag{1}$$ holds. If we consider some $z\in U$ and integrate every term of $(1)$ over $(0,z)$ we get $$ (1-\varepsilon)\frac{z^2}{2}\leq e^z-1-z \leq (1+\varepsilon)\frac{z^2}{2}\tag{2} $$ hence $$ \liminf_{z\to 0}\frac{e^z-1-z}{z^2}\geq\frac{1-\varepsilon}{2},\qquad \limsup_{z\to 0}\frac{e^z-1-z}{z^2}\leq \frac{1+\varepsilon}{2} \tag{3}$$ and since $\varepsilon$ is arbitrary, the claim follows.

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I guess we're allowed limits here, so take $e^x = \lim_{k\rightarrow \infty}(1+x/k)^k$. So for large $k$, your limit should be about the same as $$\lim_{x\rightarrow 0} \frac{(1+x/k)^k - 1 - x}{x^2} = \lim_{x\rightarrow 0} \frac{1 + {k\choose 1}\frac{x}{k} + {k \choose 2}\frac{x^2}{k^2} + \mbox{HOT} - 1 - x}{x^2} =\lim_{x\rightarrow 0} \frac{{k \choose 2}\frac{x^2}{k^2} + \mbox{HOT}}{x^2} =\frac{1}{2}.$$

If we can figure out a way to swap the two limits, we'll have a rigorous proof.

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  • $\begingroup$ Last equality is not true. It is $(k-1)/(2k)$. And this tends to $1/2$. $\endgroup$ – ajotatxe Aug 30 '16 at 15:22
  • $\begingroup$ Like I said, swap the limits. $\endgroup$ – B. Goddard Aug 30 '16 at 15:23
  • $\begingroup$ How do you swap limits without calculus? $\endgroup$ – Bernard Aug 30 '16 at 16:01
  • $\begingroup$ How do you have limits at all without calculus? "Calculus is algebra plus limits." -- Famous Wise Person. $\endgroup$ – B. Goddard Aug 30 '16 at 18:38
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    $\begingroup$ What's $\text{HOT}$ a short for? $\endgroup$ – user00000 Aug 30 '16 at 19:31
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If you allow a few basic calculus results into the game, then note

$$e^x-x - 1 = \int_0^x (e^t-1)\, dt = \int_0^x [(e^t-1)/t]t\, dt.$$

Now $(e^t-1)/t \to 1$ as $t\to 0,$ simply because the derivative of $e^x$ is $1$ at $0.$ That implies the last integral above $\approx \int_0^x t\, dt = x^2/2$ and that will lead to the result. So a few details missing, but it's not too hard to fill them in.

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