Question

find the following limit. $lim_{x
ightarrow\frac{pi}{2}}cot(5x)$

Explanation:

Step1: Recall cotangent definition

$\cot(5x)=\frac{\cos(5x)}{\sin(5x)}$

Step2: Evaluate the limit

We want to find $\lim_{x
ightarrow\frac{\pi}{5}}\cot(5x)=\lim_{x
ightarrow\frac{\pi}{5}}\frac{\cos(5x)}{\sin(5x)}$. Substitute $x = \frac{\pi}{5}$ into the expression: $\cos(5\times\frac{\pi}{5})=\cos(\pi)= - 1$ and $\sin(5\times\frac{\pi}{5})=\sin(\pi)=0$. As $x
ightarrow\frac{\pi}{5}$, the numerator approaches - 1 and the denominator approaches 0. The limit does not exist. We can consider the one - sided limits.
For $x
ightarrow\frac{\pi}{5}^+$, $\sin(5x)<0$ as $5x
ightarrow\pi^+$ and $\cos(5x)
ightarrow - 1$, so $\lim_{x
ightarrow\frac{\pi}{5}^+}\frac{\cos(5x)}{\sin(5x)}=+\infty$.
For $x
ightarrow\frac{\pi}{5}^-$, $\sin(5x)>0$ as $5x
ightarrow\pi^-$ and $\cos(5x)
ightarrow - 1$, so $\lim_{x
ightarrow\frac{\pi}{5}^-}\frac{\cos(5x)}{\sin(5x)}=-\infty$.

Answer:

The limit does not exist.