The value of limx→1 log5⁡5xlogx⁡5, is - Infinity Learn by Sri Chaitanya

A
1
B
$e$
C
-1
D
none of these

Solution:

We have,

$\begin{aligned} lim_{x \to 1} {(\log_{5} 5 x)}^{\log x} 5 = lim_{x \to 1} {(\log_{5} 5 + \log_{5} x)}^{\log_{5} x} \\ = lim_{x \to 1} {(1 + \log_{5} x)}^{\frac{1}{\log_{5} x}} = e^{lim_{x \to 1} \log_{5} x \cdot \frac{1}{\log_{5} x}} = e^{1} = e \end{aligned}$

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