What is the value of the limit f(x) = sin^2x+2 sinx x^2−4x if x a… - Vidyadip

MCQ

What is the value of the limit $\text{f}(\text{x}) = \frac{\text{sin}^2\text{x}+2\sqrt{\text{sinx}}}{\text{x}^2−4\text{x}}$ if $x$ approaches $0?$

A
$\frac{1}{\sqrt{2}}$
B
$\frac{-1}{\sqrt{2}}$
✓
$\frac{-1}{2\sqrt{2}}$
D
$\frac{-1}{\sqrt{-2}}$

✓

Answer

Correct option: C.

$\frac{-1}{2\sqrt{2}}$

This is of the form $\frac{0}{0}$,
therefore we use $L$ ’Hospital’ s rule and differentiate the numerator and
denominator.
$ =\lim_\limits{a \rightarrow b} \frac{\text{2sin}\text{x cos}+\cos\text{x}\sqrt{\text{2}}}{\text{2x}−4\text{x}}$
$= \frac{0+\sqrt{2}}{-4}$
$=\frac{-1}{2\sqrt{2}}$

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