First slide

Introduction to limits

Question

The value of $lim_{x \to 1} {(\log_{2} 2 x)}^{\log_{x} 5}$ , is

Moderate

A

5 / 2
B

$e^{\log_{2} 5}$
C

log 5 / log 2
D

$e^{\log a r 52}$

Solution

We have,
$\begin{array}{l} lim_{x \to 1} {(\log_{2} 2 x)}^{\log x^{5}} \\ = lim_{x \to 1} {(\log_{2} 2 + \log_{2} x)}^{\log x^{5}} \\ = lim_{x \to 1} {(1 + \log_{2} x)}^{1 / \log_{5} x} = e^{lim_{x \to 1} (\log_{2} x) \cdot (\log_{x} 5)} \\ = e^{lim_{x \to 1} \log_{2} 5} = e^{\log_{2} 5} \end{array}$

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