Example 1: Trig Function Limit Category 3

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{5x}$

Step 1: Try Option 1: Plug it in

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

In this example we get division by zero, which means Option 1 has failed.

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{5x}=\frac{1–\mathit{cos}\left(\mathrm{(}3\left(0\right)\mathrm{)}\right)}{5\left(0\right)}=\frac{0}{0}$

Step 2: Identify the Cosine Limit Identity format.

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

Here we have the first format.

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

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{5x}$

Step 3: Get the $x–\mathit{value}$outside the cosine function to match the $x–\mathit{value}$ inside the cosine function. Rewrite the equation to collect the Cosine Limit Identity pieces that now match, and simplify. Look to cancel common factors in the other pieces that remain.

To match the $3x$ we will need to multiply the equation by $\frac{3}{5}$. We will use $\frac{3}{5}$ because it cancels out the $\overline{)5}$ that is in front of the $x–\mathit{value}$ on the bottom of the fraction that we don’t want and replaces it with the $3$ that we do want.

Once you have multiplied your matching pieces, you will then want to group the pieces that match the Cosine Limit Identity together, and group the pieces that don’t fit the identity together.

Before you move onto the final step you will also want to look for any simplification, usually canceling of common factors, in the pieces that do not fit into your Cosine Limit Identity.

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{5x}\mathbf{\cdot }\frac{\frac{3}{5}}{\frac{3}{5}}=\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{\overline{)5}x}\mathbf{\cdot }\frac{\frac{3}{5}}{\frac{3}{\overline{)5}}}=\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{x}\mathbf{\cdot }\frac{\frac{3}{5}}{3}$

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{3x}\mathbf{\cdot }\frac{3}{5}$

Step 4: Apply the Cosine Limit Identity and Option 1: Plug It In to solve the limit.

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{3x}\mathbf{\cdot }\frac{3}{5}=0\mathbf{\cdot }\frac{3}{5}=0$

Final Result:

$\underset{x\to 0}{\lim }\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{5x}=0$

Meaning:

The overall limit as $x$ approaches $0$ of $\frac{1–\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}{5x}$ is $y=0$.