Example 1: Trig Function Limit Category 1

$\underset{x\to 0}{\mathrm{lim }}\sqrt{7+\mathit{se}{c}^2\left(\mathrm{(}x\mathrm{)}\right)}$

Step 1: Try Option 1: Plug It In

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

This is an example of why you always try Option 1: Plug It In. This equation might look intimidating at first, but $\mathit{se}{c}^2\left(\mathrm{(}0\mathrm{)}\right)=1$. So, what might at first look intimidating is not really that scary at all.

As long as you know your Unit Circle and Trig Identities .

$\underset{x\to 0}{\mathrm{lim }}\sqrt{7+\mathit{se}{c}^2\left(\mathrm{(}x\mathrm{)}\right)}=\sqrt{7+\mathit{se}{c}^2\left(\mathrm{(}0\mathrm{)}\right)}=\sqrt{7+1}=\sqrt{8}$

Final Result:

$\underset{x\to 0}{\mathrm{lim }}\sqrt{7+\mathit{se}{c}^2\left(\mathrm{(}x\mathrm{)}\right)}=\sqrt{8}$

Meaning:

The overall limit as $x$ approaches $0$ of $\sqrt{7+\mathit{se}{c}^2\left(\mathrm{(}x\mathrm{)}\right)}$ is $y=0$.