Question

The value of \(\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]\) is equal to
(A) \(e^{\sin ^{2} y}\)
(B) \(\mathrm{e}^{2 \sin y}\)
(C) \(\mathrm{e}^{|\sin y|}\)
(D) \(e^{\operatorname{cosec}^{2} y}\)

Answer 1

Ans: (A)
Hint \(:=\operatorname{Lt}_{x \rightarrow 0} \frac{1}{x}\left(\int_{y}^{a} e^{\sin ^{2} t} d t+\int_{a}^{x+y} e^{\sin ^{2} t} d t\right)\)
\(=\operatorname{Lt}_{x \rightarrow 0} \frac{1}{x} \int_{y}^{x+y} e^{\sin ^{2} t} d t\)
\(\left(\right.\) form \(\left.\frac{0}{0}\right)\)
Applying LH rule
\(=\operatorname{Lt}_{x \rightarrow 0} \frac{e^{\sin ^{2}(x+y)} \cdot 1-0}{1}=e^{\sin ^{2} y}\)

Answer 2

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Answer 3

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