_ h 0 (a + h) ^2 (a + h) - a^2 a h = - Vidyadip

MCQ

$\mathop {\lim }\limits_{h \to 0} \frac{{{{(a + h)}^2}\sin (a + h) - {a^2}\sin a}}{h} = $

A
$a\cos a + {a^2}\sin a$
B
$a\sin a + {a^2}\cos a$
✓
$2a\sin a + {a^2}\cos a$
D
$2a\cos a + {a^2}\sin a$

✓

Answer

Correct option: C.

$2a\sin a + {a^2}\cos a$

c
(c) $\frac{d}{{da}}\,[{a^2}\sin a] = 2a\sin a + {a^2}\cos a.$

Aliter : Apply $ L-$ Hospital’s rule,

$\mathop {\lim }\limits_{h \to 0} \,\frac{{{{(a + h)}^2}\sin (a + h) - {a^2}\sin a}}{h}$

$ = \mathop {\lim }\limits_{h \to 0} \,\,\frac{{2\,(a + h)\,\sin \,(a + h) + {{(a + h)}^2}\cos \,(a + h)}}{1}$

$ = 2a\,\,\sin a + {a^2}\cos \,\,a.$

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