Finding the limit $ \lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)} $

Solution 1:

Note that we can rewrite this as $$ \lim_{x\to a^+} \cos(x) \cdot \lim_{x\to a^+} \frac{\ln(x-a)}{\ln(e^{x} - e^a)} = \cos(a)\lim_{x\to a^+} \frac{\ln(x-a)}{\ln(e^{x} - e^a)} $$ In order to show that the remaining limit becomes $1$, use L'Hôpital's rule: $$ \begin{align} \lim_{x\to a^+} \frac{\ln(x-a)}{\ln(e^{x} - e^a)} &= \lim_{x\to a^+} \frac{1/(x-a)}{1/(e^{x} - e^a)\cdot e^x}\\ &=\lim_{x\to a^+} e^{-x} \lim_{x\to a^+} \frac{e^x - e^a}{x-a}\\ &= e^{-a} \lim_{x\to a^+} \frac{e^x - e^a}{x-a} \end{align} $$ Using L'Hôpital's rule once again gives us $$ \begin{align} \lim_{x\to a^+} \frac{\ln(x-a)}{\ln(e^{x} - e^a)} &= e^{-a} \lim_{x\to a^+} \frac{e^x}{1} = e^{-a}e^a = 1 \end{align} $$ Thus, the limit of our original function is $\cos(a)$.

Solution 2:

We put $x = a + h$ so that when $x \to a+$ then $h \to 0+$. We then have

\begin{align} L &= \lim_{x \to a+}\frac{\cos x \ln(x - a)}{\ln(e^{x} - e^{a})}\notag\\ &= \lim_{h \to 0+}\cos(a + h)\frac{\ln h}{\ln(e^{a + h} - e^{a})}\notag\\ &= \lim_{h \to 0+}\cos a \frac{\ln h}{\ln\{e^{a}(e^{h} - 1)\}}\tag{1}\\ &= \cos a\lim_{h \to 0+}\frac{\ln h}{a + \ln(e^{h} - 1)}\notag\\ &= \cos a\lim_{h \to 0+}\dfrac{\ln h}{a + \ln\left(h\cdot\dfrac{e^{h} - 1}{h}\right)}\notag\\ &= \cos a\lim_{h \to 0+}\dfrac{\ln h}{a + \ln h + \ln\left(\dfrac{e^{h} - 1}{h}\right)}\notag\\ &= \cos a\lim_{h \to 0+}\dfrac{1}{\dfrac{a}{\ln h} + 1 + \dfrac{1}{\ln h}\cdot\ln\left(\dfrac{e^{h} - 1}{h}\right)}\notag\\ &= \cos a\cdot\frac{1}{0 + 1 + 0\cdot 0}\tag{2}\\ &= \cos a\notag \end{align}

At the end we use the fact that as $h \to 0+$, $\ln h \to -\infty$ so that $a/\ln h \to 0$ and $(e^{h} - 1)/h \to 1$ so that $\ln((e^{h} - 1)/h) \to 0$ and $1/\ln h \to 0$.

As I have pointed out elsewhere in this site, we don't need L'Hospital or series expansions for most of the limit problems.

Note: In step marked $(1)$ we assume that $\cos a\neq 0$ in order for the product rule of limits to work. And finally in step $(2)$ we see that the limit of the other factor apart from $\cos a$ is $1$ so that the condition $\cos a\neq 0$ is no longer necessary and result holds for all values of $a$. Thanks to user @LearningMath for pointing this out via comments.

Solution 3:

Since for every $x \in \mathbb{R}$ we have $$ e^x-e^a=e^a(e^{x-a}-1)=e^a\left[x-a+\frac{(x-a)^2}{2!}+\frac{(x-a)^3}{3!}+\ldots\right]=(x-a)R(x;a), $$ where $$ R(x;a)=e^a\left[1+\frac{x-a}{2!}+\frac{(x-a)^2}{3!}+\ldots\right] $$ it follows that for every $x>a$ we have \begin{eqnarray} f(x):&=&\frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)}=\cos(x)\frac{\ln(x-a)}{\ln(x-a)+\ln R(x;a)}\\ &=&\frac{\cos(x)}{1+\frac{\ln R(x;a)}{\ln(x-a)}} \end{eqnarray} We have $$ \lim_{x\to a^+}\frac{1}{\ln(x-a)}=0,\ \lim_{x\to a^+}\ln R(x;a)=\ln e^a=a, $$ and we conclude that $$ \lim_{x\to a+}\frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)}=\frac{\cos a}{1+0\cdot a}=\cos a. $$