Calculus Examples

$lim_{x \to 0} \frac{\tan (9 x)}{\sin (3 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (9 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (9 x)}{\cos (9 x)}}{\sin (3 x)}$

Step 1.2

Rewrite $\frac{\frac{\sin (9 x)}{\cos (9 x)}}{\sin (3 x)}$ as a product.

$lim_{x \to 0} \frac{\sin (9 x)}{\cos (9 x)} \cdot \frac{1}{\sin (3 x)}$

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{\sin (9 x)}{\cos (9 x)}$ to $\tan (9 x)$ .

$lim_{x \to 0} \tan (9 x) \frac{1}{\sin (3 x)}$

Step 1.3.2

Convert from $\frac{1}{\sin (3 x)}$ to $\csc (3 x)$ .

$lim_{x \to 0} \tan (9 x) \csc (3 x)$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \tan (9 x) \csc (3 x)$

Step 3

Make a table to show the behavior of the function $\tan (9 x) \csc (3 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \tan (9 x) \csc (3 x) \\ - 0.1 & 4.26420322 \\ - 0.01 & 3.00857759 \\ - 0.001 & 3.0000855 \\ - 0.0001 & 3.00000085 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $3$ . Thus, the limit of $\tan (9 x) \csc (3 x)$ as $x$ approaches $0$ from the left is $3$ .

$3$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} \tan (9 x) \csc (3 x)$

Step 6

Make a table to show the behavior of the function $\tan (9 x) \csc (3 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \tan (9 x) \csc (3 x) \\ 0.1 & 4.26420322 \\ 0.01 & 3.00857759 \\ 0.001 & 3.0000855 \\ 0.0001 & 3.00000085 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $3$ . Thus, the limit of $\tan (9 x) \csc (3 x)$ as $x$ approaches $0$ from the right is $3$ .

$3$