Example 2: Trig Function Limit Category 1

$\underset{x\to 0}{\mathrm{lim }}\mathit{cot}\left(x\right)\mathit{sin}\left(x\right)$

Step 1: Try Option 1: Plug It In

Always try plugging in the $x–\mathit{value}$ you are heading towards.

In this example Option 1 fails because cotangent is not defined (undefined) at $x=0$.

$\underset{x\to 0}{\mathrm{lim }}\mathit{cot}\left(x\right)\mathit{sin}\left(x\right)=\mathit{cot}\left(0\right)\mathit{sin}\left(0\right)=\left(\mathit{Undefined}\right)\left(0\right)=\mathit{Undefined}$

Step 2: Rewrite all trig functions into their sine and cosine identities, and look to cancel out common factors.

Here cotangent can be rewritten using its sine and cosine identity.

After the rewrite you can see that you have some sine equations that will cancel. Conveniently this leaves us with a new easy to solve limit.

$\underset{x\to 0}{\mathrm{lim }}\mathit{cot}\left(x\right)\mathit{sin}\left(x\right)=\underset{x\to 0}{\mathrm{lim }}\frac{\cos \mathrm{(}x\mathrm{)}}{\mathrm{sin }\left(\mathrm{x}\right)}\mathit{sin}\left(x\right)=\underset{x\to 0}{\mathrm{lim }}\frac{\cos \mathrm{(}x\mathrm{)}}{\overline{)\mathit{sin}\left(x\right)}}\overline{)\mathit{sin}\left(x\right)}=\underset{x\to 0}{\mathrm{lim }}\mathit{cos}\left(x\right)$

Step 3: Back to Option 1: Plug It In

$\underset{x\to 0}{\mathrm{lim }}\mathit{cos}\left(x\right)=\mathit{cos}\left(0\right)=\mathit{1}$

Final Result:

$\underset{x\to 0}{\mathrm{lim }}\mathit{cot}\left(x\right)\mathit{sin}\left(x\right)=1$

Meaning:

The overall limit as $x$ approaches $0$ of $\mathit{cot}\left(x\right)\mathit{sin}\left(x\right)$ is $y=1$.