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  • <img alt="\mathrm{\text { Evaluate } \quad \lim _{x \rightarrow 0} x \cot (6 x)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Ctext%20%7B%20Evaluate%20%7D%20%5Cquad%20%5Clim%

\mathrm{\text { Evaluate } \quad \lim _{x \rightarrow 0} x \cot (6 x)}

Option: 1

1 / 6


Option: 2

2 / 3


Option: 3

0


Option: 4

2 / 5


Answers (1)

best_answer

Convert  \mathrm{\cot 6 x} to  \mathrm{\tan 6 x } and multiply and divide by \mathrm{6} : 

                                                                      \mathrm{\begin{aligned} & \lim _{x \rightarrow 0} x \cot (6 x) \\ = & \lim _{x \rightarrow 0} \frac{x}{\tan (6 x)} \\ = & \lim _{x \rightarrow 0} \frac{6 x}{6 \tan (6 x)} \end{aligned}}

We know that 

                                                                         \mathrm{\lim _{x \rightarrow 0} \frac{\tan x}{x}=1 .}

Therefore, 

                                                    \mathrm{\lim _{x \rightarrow 0} \frac{6 x}{6 \tan (6 x)}=\frac{1}{6} \lim _{x \rightarrow 0} \cdot \frac{6 x}{\tan (6 x)}=\frac{1}{6}}

Posted by

Shailly goel

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