_ x 4 [ x^ 3/2 - 8 x - 4 ] = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 4} \left[ {\frac{{{x^{3/2}} - 8}}{{x - 4}}} \right] = $

A
$3/2$
✓
$3$
C
$2/3$
D
$1/3$

✓

Answer

Correct option: B.

$3$

b
(b) $y = \mathop {\lim }\limits_{x \to 4} \left[ {\frac{{{x^{3/2}} - 8}}{{x - 4}}} \right]$$ = \mathop {\lim }\limits_{x \to 4} \,\left[ {\frac{{{{({x^{1/2}})}^3} - {{(2)}^3}}}{{(\sqrt x - 2)(\sqrt x + 2)}}} \right]$

==> $y = \mathop {\lim }\limits_{x \to 4} \frac{{({x^{1/2}} - 2)(x + 4 + 2\sqrt x )}}{{(\sqrt x - 2)(\sqrt x + 2)}}$

==> $y = \mathop {\lim }\limits_{x \to 4} \frac{{(x + 4 + 2\sqrt x )}}{{(\sqrt x + 2)}}$$ = \frac{{4 + 4 + 2\sqrt 4 }}{{\sqrt 4 + 2}}$$ = \frac{{12}}{4} = 3$.

Trick : Applying $ L-$ Hospital’s rule, we get

$\mathop {\lim }\limits_{x \to 4} \frac{{\frac{3}{2}{x^{1/2}}}}{1}$$ = \frac{3}{2}{(4)^{1/2}} = 3.$

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