First slide

Introduction to limits

Question

The value of $lim_{x \to 1} {(\log_{3} 3 x)}^{\log_{x} 3},$ is

Moderate

A

$e$
B

$\frac{1}{e}$
C

1
D

$- \frac{1}{e}$

Solution

We have,
$\begin{array}{l} lim_{x \to 1} {(\log_{3} 3 x)}^{\log_{x} 3} \\ = lim_{x \to 1} {(\log_{3} 3 + \log_{3} x)}^{\log_{x} 3} \\ = lim_{x \to 1} {(1 + \log_{3} x)}^{\frac{1}{\log_{3} x}} = e^{lim_{x \to 1} \frac{\log_{3} x}{\log_{3} x}} = e \end{array}$

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