Calculus Examples

$lim_{x \to 0} \frac{\tan (2 x)}{\frac{1}{\csc (4 x)}}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\csc (4 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\tan (2 x)}{\frac{1}{\frac{1}{\sin (4 x)}}}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (4 x)}$ .

$lim_{x \to 0} \frac{\tan (2 x)}{1 \sin (4 x)}$

Step 1.3

Multiply $\sin (4 x)$ by $1$ .

$lim_{x \to 0} \frac{\tan (2 x)}{\sin (4 x)}$

Step 1.4

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (2 x)}{\cos (2 x)}}{\sin (4 x)}$

Step 1.5

Rewrite $\frac{\frac{\sin (2 x)}{\cos (2 x)}}{\sin (4 x)}$ as a product.

$lim_{x \to 0} \frac{\sin (2 x)}{\cos (2 x)} \cdot \frac{1}{\sin (4 x)}$

Step 1.6

Simplify.

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Step 1.6.1

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$lim_{x \to 0} \tan (2 x) \frac{1}{\sin (4 x)}$

Step 1.6.2

Convert from $\frac{1}{\sin (4 x)}$ to $\csc (4 x)$ .

$lim_{x \to 0} \tan (2 x) \csc (4 x)$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \tan (2 x) \csc (4 x)$

Step 3

Make a table to show the behavior of the function $\tan (2 x) \csc (4 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \tan (2 x) \csc (4 x) \\ - 0.1 & 0.52054567 \\ - 0.01 & 0.50020005 \\ - 0.001 & 0.500002 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $0.5$ . Thus, the limit of $\tan (2 x) \csc (4 x)$ as $x$ approaches $0$ from the left is $0.5$ .

$0.5$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} \tan (2 x) \csc (4 x)$

Step 6

Make a table to show the behavior of the function $\tan (2 x) \csc (4 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \tan (2 x) \csc (4 x) \\ 0.1 & 0.52054567 \\ 0.01 & 0.50020005 \\ 0.001 & 0.500002 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $0.5$ . Thus, the limit of $\tan (2 x) \csc (4 x)$ as $x$ approaches $0$ from the right is $0.5$ .

$0.5$