MCQ
The value of $\mathop {\lim }\limits_{x \to 0} \frac{{\log [1 + {x^3}]}}{{{{\sin }^3}x}} = $
  • A
    $0$
  • $1$
  • C
    $3$
  • D
    None of these

Answer

Correct option: B.
$1$
b
(b) $\mathop {\lim }\limits_{x \to 0} \frac{{\log (1 + {x^3})}}{{{{\sin }^3}x}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{3{x^2}/(1 + {x^3})}}{{3{{\sin }^2}x\cos x}}$

[By using $ L-$  Hospital rule]

$ = \mathop {\lim }\limits_{x \to 0} \left[ {\frac{1}{{1 + {x^3}}}{{\left( {\frac{x}{{\sin x}}} \right)}^2}.\frac{1}{{\cos x}}} \right]$

$ = \frac{1}{{1 + 0}}.{(1)^2}.\frac{1}{1} = 1$.

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