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Question: \(\lim _ { x \rightarrow 0 }\) \(\frac{\int_{0}^{x^{2}}{\tan\sqrt{t}dt}}{x^{3}}\) is equal to -...

lim⁡x→0\lim _ { x \rightarrow 0 } ∫0x2tan⁡tdtx3\frac{\int_{0}^{x^{2}}{\tan\sqrt{t}dt}}{x^{3}} is equal to -

A

23\frac{2}{3}

B

32\frac{3}{2}

C

1

D

None of these

Answer

None of these

Explanation

Solution

lim⁡x→0\lim_{x \rightarrow 0} ∫0x2tan⁡tdt/x3\int_{0}^{x^{2}}{\tan\sqrt{t}dt}/x^{3} 0/0 form

App.L-HŽlim⁡x→0\lim _ { x \rightarrow 0 } tan⁡x2.2x–0/3x2\tan\sqrt{x^{2}}.2x–0/3x^{2}

Ž 2/3 lim⁡x→0\lim_{x \rightarrow 0} tan |x|/x

R.H.L. 2/3lim⁡x→0\lim_{x \rightarrow 0}tanx/x= 2/3 limit does not exist.

L.H.L 2/3 lim⁡x→0\lim_{x \rightarrow 0} tan(–x)/x= – 2/3