Example 1: Trig Function Limit Category 2

$\underset{x\to 0}{\lim }\frac{\mathit{tan}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}$

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.

$\underset{x\to 0}{\lim }\frac{\mathit{tan}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}=\frac{\mathit{tan}\left(\mathrm{(}3\left(0\right)\mathrm{)}\right)}{\mathit{sin}\left(\mathrm{(}8\left(0\right)\mathrm{)}\right)}=\frac{0}{0}$

Step 2: Rewrite all trig functions into their sine and cosine identities.

$\underset{x\to 0}{\lim }\frac{\mathit{tan}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}=\underset{\mathrm{x}\to 0}{\lim }\frac{\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{1}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}=$

Step 3: Identify all of your different sine functions.

In this example we have two different sine functions that we will need to match,

$\mathit{sin}\left(3x\right)$and $\mathit{sin}\left(8x\right)$.

It really is like having two problems in one here, and we will be using both versions of the sine limit identity.

$\underset{x\to 0}{\lim }\frac{\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{1}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}=$

Step 4: Get the $x–\mathit{value}$outside the sine function to match the $x–\mathit{value}$ inside the sine function, rewrite the equation to collect the Sine 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 $\frac{3x}{3x}$.

To match the $8x$ we will need to multiply the equation by $\frac{8x}{8x}$.

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

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 Sine Limit Identity.

$\underset{x\to 0}{\lim }\frac{\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{3x}{3x}\mathbf{\cdot }\frac{1}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}\mathit{\cdot }\frac{8x}{8x}=$

$\underset{x\to 0}{\lim }\frac{\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}{3x}\mathbf{\cdot }\frac{8x}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}\mathbf{\cdot }\frac{1}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{3x}{8x}=$

$\underset{x\to 0}{\lim }\frac{\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}{3x}\mathbf{\cdot }\frac{8x}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}\mathbf{\cdot }\frac{1}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{3\overline{)x}}{8\overline{)x}}=$

$\underset{x\to 0}{\lim }\frac{\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}{3x}\mathbf{\cdot }\frac{8x}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}\mathbf{\cdot }\frac{1}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{3}{8}=$

Step 5: Apply the Sine Limit Identity and Option 1: Plug It In to solve the limit.

$\underset{x\to 0}{\lim }\frac{\overline{)\mathit{sin}\left(\mathrm{(}3x\mathrm{)}\right)}}{\overline{)3x}}\mathbf{\cdot }\frac{\overline{)8x}}{\overline{)\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}}\mathbf{\cdot }\frac{1}{\mathit{cos}\left(\mathrm{(}3x\mathrm{)}\right)}\mathbf{\cdot }\frac{3}{8}=1\mathbf{\cdot }1\mathbf{\cdot }\frac{1}{\mathit{cos}\left(\mathrm{(}3\left(0\right)\mathrm{)}\right)}\mathbf{\cdot }\frac{3}{8}=\frac{3}{8}$

Final Result:

$\underset{x\to 0}{\lim }\frac{\mathit{tan}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}=\frac{3}{8}$

Meaning:

The overall limit as $x$ approaches $0$ of $\frac{\mathit{tan}\left(\mathrm{(}3x\mathrm{)}\right)}{\mathit{sin}\left(\mathrm{(}8x\mathrm{)}\right)}$ is $y=\frac{3}{8}$.