_ x 0 , , _e (1 + x) 3^x - 1 = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \,\,\frac{{{{\log }_e}(1 + x)}}{{{3^x} - 1}} = $

A
${\log _e}3$
B
$0$
C
$1$
✓
${\log _3}e$

✓

Answer

Correct option: D.

${\log _3}e$

d
(d) $\mathop {\lim }\limits_{x \to 0} \frac{{{{\log }_e}(1 + x)}}{{{3^x} - 1}}$, $\left( {\frac{0}{0}\,{\rm{ form}}} \right)$

Using $ L-$ Hospital’s rule,

$\mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{{1 + x}}}}{{{3^x}{{\log }_e}3}} = \frac{1}{{{{\log }_e}3}} = {\log _3}e$.

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