lim-x-0-0-x-1-sin-t-dt-x-a-then-a-2-1-

Question Number 169743 by naka3546 last updated on 07/May/22

\lim_{x \to 0} \frac{\int_{0} \overset{x}{} \sqrt{1 + \sin t} d t}{x} = a,

t h e n a^{2} - 1 = \dots ?

Answered by Mathspace last updated on 07/May/22

b y h o s i t a l {l i m}_{x \to 0} \frac{\int_{0}^{x} \sqrt{1 + s i n t} d t}{x}

= {l i m}_{x \to 0} \frac{\sqrt{1 + s i n x}}{1} = 1 \Rightarrow a = 1 \Rightarrow

a^{2} - 1 = 0

Commented by naka3546 last updated on 07/May/22

Thank you, s i r .

Answered by cortano1 last updated on 08/May/22

2^{n d} w a y

L = \lim_{x \to 0} \frac{\int_{0}^{x} \sqrt{1 + \sin t} d t}{x}

= \lim_{x \to 0} \frac{\int_{0}^{x} (\sin \frac{1}{2} t + \cos \frac{1}{2} t) d t}{x}

= \lim_{x \to 0} \frac{- 2 \cos \frac{1}{2} x + 2 \sin \frac{1}{2} x + 2}{x}

= \lim_{x \to 0} \frac{2 (\sin \frac{1}{2} x + 2 \sin^{2} \frac{1}{4} x)}{x}

= 2 (\frac{1}{2} + 2 . \frac{1}{4} . 0) = 1 = a

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