The value of lim x → 0 1 − c o s 3 x x s i n x c o s x Options: A . &nbsp 2/5 B . &nbsp 3/5 C . &nbsp 3/2 D . &nbsp 3/4

Answer: Option C : C lim x → 0 1 − c o s 3 x x s i n x c o s x = lim x → 0 ( 1 − c o s x ) ( 1 + c o s x + c o s 2 x ) x s i n x c o s x = lim x → 0 2 s i n 2 ( x 2 ) 2 s i n ( x 2 ) c o s ( x 2 ) . x × ( 1 + c o s x + c o s 2 x ) c o s x = lim x → 0 s i n ( x 2 ) 2 ( x 2 ) × 1 + c o s x + c o s 2 x c o s ( x 2 ) c o s x = 1 2 × 3 = 3 2

The value oflimx→01−cos3xxsin xcosx

Discussion Forum : Limits Continuity And Differentiability

Question -

$The value of {lim}_{x \to 0} \frac{1 - c o s^{3} x}{x s i n x c o s x}$

Options:

A . 2/5

B . 3/5

C . 3/2

D . 3/4

Answer: Option C
:
C

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

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Discussion Forum : Limits Continuity And Differentiability

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