The limit is $$\lim_{x\to0}\left(\frac{\ln(1+x)-\ln(1-x)}{x}\right)$$ Please resolve without using l'hôpital's rule, haven't made it that far yet. Thanks in advance.
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Have you learned how to find limit using definition of derivative yet? if so, you can write it like this:
$\displaystyle \lim_{x \to 0}\dfrac{\ln(1+x) - \ln(1-x)}{x} = \displaystyle \lim_{x \to 0}\left(\dfrac{\ln(1+x)}{x} - \dfrac{\ln(1-x)}{x}\right)= \dfrac{d}{dx}|_{x = 0} \ln(1+x) - \dfrac{d}{dx}|_{x=0} \ln(1-x)=1 - (-1) = 2$
DeepSea
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Is this not L'hopitals? – John Lou May 24 '17 at 01:27
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4@JohnLou. This is not L'Hopital's Rule. It's applying the definition of derivative to calculate a limit. – sharding4 May 24 '17 at 01:28
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No, we haven't even started with derivatives yet. – Alvaro Morales May 24 '17 at 01:29
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It seems really similar to L'hopitals, though, as if it were a rudimentary version. – John Lou May 24 '17 at 01:30
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Completely irrelevant comment: Did anyone ever think L'Hopital's was spelled L'Hospital's before? I did for some reason – Eric Lee May 24 '17 at 01:33
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@EricLee L'Hopitals is spelled with an accent circumflex (over the o) in the original French. That usually indicates a missing "s" when converting to English, so it is often spelled both ways. For example, Forest in French is "foret" with an accent over the "s." – John Lou May 24 '17 at 01:39
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The more you know :D Thank you! – Eric Lee May 24 '17 at 01:39
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2@JohnLou It is not LHR. It bears a superficial resemblance to it, but it's elementary, and LHR isn't, depending as it does on values of derivatives away from the point in question as well as the Cauchy generalized MVT. – zhw. May 24 '17 at 22:38
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Let $\displaystyle y=\ln\left(\frac{1+x}{1-x}\right)$. Then
$$e^y=\frac{1+x}{1-x}$$
$$x=\frac{e^y-1}{e^y+1}$$
As $x\to 0$, $y\to0$.
$$\lim_{x\to 0}\frac{\ln(1+x)-\ln(1-x)}{x}=\lim_{y\to0}\frac{y(e^y+1)}{e^y-1}=\lim_{y\to0}\frac{y}{e^y-1}\cdot\lim_{y\to0}(e^y+1)=2$$
CY Aries
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1How did you evaluate $$\lim_{y\to0}\frac{y}{e^y-1}$$ without L'Hopital? – John Doe May 24 '17 at 01:36
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1@Cerkal No. There are many ways to define the constant $e$. Of course, those definitions are equivalent. We can define $e$ as the number such that $\lim_{h\to0}\frac{e^h-1}{h}=1$. https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function – CY Aries May 24 '17 at 02:21
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There is a very important and useful Taylor series $$\log \left(\frac{1+x}{1-x}\right)=2\sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1}=2 x+\frac{2 x^3}{3}+O\left(x^5\right)$$ which makes $$\frac 1x\log \left(\frac{1+x}{1-x}\right)=2+\frac{2 x^2}{3}+O\left(x^4\right)$$ which shows the limit and also how it is approached.
Claude Leibovici
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I still remember the time when I first met this series (20 years ago) and used it to calculate $\log 3$ by putting $x=1/2$. Much better than the usual series for $\log(1+x)$ because it converges faster and has all terms positive. +1 – Paramanand Singh May 24 '17 at 18:56