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Question Number 106217 by bobhans last updated on 03/Aug/20

If f (x) = \frac{\sin x - x \cos x}{x^{2}} and f (0) = 0 when x

= 0, what will be the value of \lim_{t \to 0} \frac{\underset{0}{\int^{t}} f (x) dx}{t^{2}}

Answered by john santu last updated on 03/Aug/20

L^{'} Hopital : \lim_{t \to 0} \frac{\underset{0}{\int^{t}} f (x) dx}{t^{2}} =

\lim_{t \to 0} \frac{f (t)}{2 t} = \lim_{t \to 0} \frac{\sin t - t . \cos t}{2 t . t^{2}}

= \lim_{t \to 0} \frac{(t - \frac{t^{3}}{6}) - t (1 - \frac{t^{2}}{2})}{{2 t}^{3}}

= \lim_{t \to 0} \frac{\frac{t^{3}}{2} - \frac{t^{3}}{6}}{{2 t}^{3}} = \frac{2}{12} = \frac{1}{6} . (JS)