# Compute the limit: $\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)$

Does anyone know how o compute the following limit?

$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)$$

I tried to split it up as follows:

$\lim_{x\to \infty}\left[x\ \log(x+c)-x\ \log(x-c)\right]$ but still this is indeterminate form of type $\infty-\infty$. Any thoughts please?

• $$\frac{x+c}{x-c} = \frac{1 + \frac{c}{x}}{1-\frac{c}{x}}$$ – Daniel Fischer Nov 16 '14 at 23:47

Using L'Hospital:

$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)=\lim_{x\to \infty}\frac{\log\left(\frac{x+c}{x-c}\right)}{\frac 1x}\underbrace{=}_{\mathrm{L'Hospital}} \lim_{x\to \infty} \frac{\frac{x-c}{x+c}\cdot \frac{-2c}{(x-c)^2}}{-\frac {1}{x^2}}=\lim_{x\to \infty} \frac{x-c}{x+c}\cdot \frac{2cx^2}{(x-c)^2}=2c$$

Without using L'Hospital:

$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)=\lim_{x\to \infty} \log\left(1+\frac{2c}{x-c}\right)^x=\log \lim_{x\to \infty}\left(1+\frac{2c}{x-c}\right)^x\\=\log \lim_{x\to \infty}\left(\left(1+\frac{2c}{x-c}\right)^{x-c}\right)^{\frac{x}{x-c}}=\log e^{2c}=2c.$$

• In the second expression in the "without" section, that should be $1+\frac{2c}{x-c}$, not minus. That will make the second answer the same as the first. – Rory Daulton Nov 17 '14 at 0:02
• @RoryDaulton Fixed. Thank you for noticing it. – mfl Nov 17 '14 at 0:12

We have:$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right) = \lim_{x\to \infty}\ \log\left(\frac{x+c}{x-c}\right)^x = \log \lim_{x \to +\infty}\left(\frac{1+\frac{c}{x}}{1-\frac{c}{x}}\right)^x = \log \lim_{x \to +\infty} \frac{\left(1+\frac{c}{x}\right)^x}{\left(1-\frac{c}{x}\right)^x}$$

Using the fundamental limit for $e$: $$= \log \frac{e^c}{e^{-c}} = \log e^{2c} = 2c.$$

Let $u = \dfrac{c}{x}$, then:

Note that: $x\log\left(\dfrac{x+c}{x-c}\right) = c\dfrac{\log\left(\dfrac{1+u}{1-u}\right)}{u} = c\left(\dfrac{\log(1+u)}{u} - \dfrac{\log(1-u)}{u}\right) \to c(1-(-1)) = 2c$ since $x \to \infty \iff u \to 0$.

Hint; Let $t=\frac1x$${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$