Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 38

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Determine the $\displaystyle \lim_{x \to a^+} \frac{\cos x \ln (x - a)}{\ln \left( e^x - e^a\right)}$. Use L'Hospital's Rule where appropriate. Use some Elementary method if posible. If L'Hospitals Rule doesn't apply. Explain why.

\begin{aligned} \lim_{x \to a^+} \frac{\cos x \ln (x - a)}{\ln \left( e^x - e^a\right)} &= \lim_{x \to a^+} \cos x \cdot \lim_{x \to a^+} \left[ \frac{\ln(x-a)}{\ln(e^x - e^a)} \right]\\ \\ &= \cos a \cdot \lim_{x \to a^+} \left[ \frac{\ln(x-a)}{\ln \left( e^x - e^a\right)} \right] \end{aligned}

By applying L'Hospital's Rule..

\begin{aligned} &= \cos a \cdot \lim_{x \to a^+} \left[ \frac{\frac{1}{x-a}}{\frac{e^x}{e^x - e^a}} \right]\\ \\ &= \cos a \cdot \lim_{x \to a^+} \left( \frac{e^x - e^a}{e^x(x-a)} \right)\\ \\ &= \cos a \left[ \lim_{x \to a^+}\left( \frac{1}{e^x} \right) \cdot \lim_{x \to a^+} \left( \frac{e^x - e^a}{x -a} \right)\right]\\ \\ &= \frac{\cos a}{e^a} \cdot \lim_{x \to a^+} \left( \frac{e^x - e^a}{x - a} \right) \end{aligned}

Again, by applying L'Hospital's Rule...

\begin{aligned} &= \frac{\cos a}{e^a} \cdot \lim_{x \to a^+} \left( \frac{e^x}{1} \right)\\ \\ &= \frac{\cos a}{e^a} \cdot \lim_{x \to a^+} e^x\\ \\ &= \frac{\cos a}{e^a} \cdot e^a \quad = \cos a \end{aligned}