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Question

Mathematics Question on Derivatives

The value of lim⁡x→0cot⁡4xcosec3x \displaystyle\lim _{x \rightarrow 0} \frac{\cot 4 x}{\text{cosec} 3 x} is equal to

A

43\frac{4}{3}

B

34\frac{3}{4}

C

23\frac{2}{3}

D

32\frac{3}{2}

Answer

34\frac{3}{4}

Explanation

Solution

Given, lim⁡x→0cot⁡4xcosec⁡3x\displaystyle\lim _{x \rightarrow 0} \frac{\cot 4 x}{\operatorname{cosec} 3 x}
=lim⁡x→0(sin⁡3xtan⁡4x)=\displaystyle\lim _{x \rightarrow 0}\left(\frac{\sin 3 x}{\tan 4 x}\right)
lim⁡x→0sin⁡3x3x×3xtan⁡4x4x×4x\displaystyle\lim _{x \rightarrow 0} \frac{\frac{\sin 3 x}{3 x} \times 3 x}{\frac{\tan 4 x}{4 x} \times 4 x}
=lim⁡x→03x4x=34=\displaystyle\lim _{x \rightarrow 0} \frac{3 x}{4 x}=\frac{3}{4}