Calculus Examples

$lim_{x \to 0} \frac{\tan (7 x)}{\sin (5 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (7 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (7 x)}{\cos (7 x)}}{\sin (5 x)}$

Step 1.2

Rewrite $\frac{\frac{\sin (7 x)}{\cos (7 x)}}{\sin (5 x)}$ as a product.

$lim_{x \to 0} \frac{\sin (7 x)}{\cos (7 x)} \cdot \frac{1}{\sin (5 x)}$

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{\sin (7 x)}{\cos (7 x)}$ to $\tan (7 x)$ .

$lim_{x \to 0} \tan (7 x) \frac{1}{\sin (5 x)}$

Step 1.3.2

Convert from $\frac{1}{\sin (5 x)}$ to $\csc (5 x)$ .

$lim_{x \to 0} \tan (7 x) \csc (5 x)$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \tan (7 x) \csc (5 x)$

Step 3

Make a table to show the behavior of the function $\tan (7 x) \csc (5 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \tan (7 x) \csc (5 x) \\ - 0.1 & 1.75687007 \\ - 0.01 & 1.40287561 \\ - 0.001 & 1.4000287 \\ - 0.0001 & 1.40000028 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $1.4$ . Thus, the limit of $\tan (7 x) \csc (5 x)$ as $x$ approaches $0$ from the left is $1.4$ .

$1.4$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} \tan (7 x) \csc (5 x)$

Step 6

Make a table to show the behavior of the function $\tan (7 x) \csc (5 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \tan (7 x) \csc (5 x) \\ 0.1 & 1.75687007 \\ 0.01 & 1.40287561 \\ 0.001 & 1.4000287 \\ 0.0001 & 1.40000028 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $1.4$ . Thus, the limit of $\tan (7 x) \csc (5 x)$ as $x$ approaches $0$ from the right is $1.4$ .

$1.4$