The value of lim x → 0 3 √ 1 + s i n x − 3 √ 1 − s i n x x i s Options: A . &nbsp 2/3 B . &nbsp -2/3 C . &nbsp 3/2 D . &nbsp -3/2

Answer: Option A : A lim x → 0 3 √ 1 + s i n x − 3 √ 1 − s i n x x = lim x → 0 ( 1 + s i n x ) − ( 1 − s i n s ) ( 1 + s i n x ) 2 / 3 + ( 1 + s i n x ) 2 / 3 + ( 1 − s i n x ) 2 / 3 ( 1 − s i n x ) 2 / 3 × 1 x = lim x → 0 2 ( 1 + s i n x ) 2 / 3 + ( 1 + s i n x ) 1 / 3 + ( 1 − s i n x ) 1 / 3 + ( 1 − s i n x ) 2 / 3 × s i n ( x ) x = 2 3 .1 = 2 3

The value oflimx→03√1+sinx−3√1−sinxxis

Discussion Forum : Limits Continuity And Differentiability

Question -

$The value of {lim}_{x \to 0} \frac{\sqrt[3]{1 + s i n x} - \sqrt[3]{1 - s i n x}}{x} i s$

Options:

A . 2/3

B . -2/3

C . 3/2

D . -3/2

Answer: Option A
:
A

${lim}_{x \to 0} \frac{\sqrt[3]{1 + s i n x} - \sqrt[3]{1 - s i n x}}{x} = {lim}_{x \to 0} \frac{(1 + s i n x) - (1 - s i n s)}{(1 + s i n x)^{2 / 3} + (1 + s i n x)^{2 / 3} + (1 - s i n x)^{2 / 3} (1 - s i n x)^{2 / 3}} \times \frac{1}{x} = {lim}_{x \to 0} \frac{2}{(1 + s i n x)^{2 / 3} + (1 + s i n x)^{1 / 3} + (1 - s i n x)^{1 / 3} + (1 - s i n x)^{2 / 3}} \times \frac{s i n (x)}{x}$
= $\frac{2}{3} .1 = \frac{2}{3}$

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Discussion Forum : Limits Continuity And Differentiability

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