Question

If f(x) is the primitive of $\frac{\sin\sqrt[3]{x}\log(1 + 3x)}{(\tan^{–1}\sqrt{x})^{2}(e^{\sqrt[3]{x}}–1)}$ (x ¹ 0), then $\lim_{x \rightarrow 0}$ f ' (x) is –

• A. 0
• B. 3/5
• C. 5/3
• D. None

Answer 1

Answer: (D)

None

Answer 2

Q f(x) = $\int_{}^{}\frac{\sin x^{1/3}\log(1 + 3x)}{(\tan^{–1}\sqrt{x})^{2}(e^{x^{1/3}}–1)}$

\ $\lim _ { x \rightarrow 0 }$ f'(x)

̃ $\lim _ { x \rightarrow 0 }$ $\frac{\left( \frac{\sin x^{1/3}}{x^{1/3}} \right)x^{1/3}\left\lbrack \frac{\cos(1 + 3x)}{3x} \right\rbrack(3x)}{\left( \frac{\tan^{–1}\sqrt{x}}{\sqrt{x}} \right)^{2}.x.\left( \frac{e^{x^{1/3}}–1}{x^{1/3}} \right).x^{1/3}}$= 3