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  4. underset x arrow 0 textLim (∫ limits0x sin t2 d t/x(1- cos x)) equals

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Q. $\underset {x \rightarrow 0}{\text{Lim}} \frac{\int\limits_0^x \sin t^2 d t}{x(1-\cos x)}$ equals

Integrals

Solution:

$\underset{x \rightarrow 0}{\text{Lim}}\frac{\int\limits_0^x \sin t^2 d t}{x^3 \frac{(1-\cos x)}{x^2}} $ (using $\underset{x \rightarrow 0}{\text{Lim}}\frac{1-\cos x}{x^2}=\frac{1}{2}$ )
$=2 \underset{x \rightarrow 0}{\text{Lim}} \frac{\int\limits_0^x \sin t^2 d t}{x^3}$ (Using L'Hospital Rule)
$=2 \underset{x \rightarrow 0}{\text{Lim}} \frac{\sin x^2}{3 x^2}=\frac{2}{3}$