Calculus Examples

$lim_{x \to 0} \frac{5}{2 x \csc (5 x)}$

Step 1

Move the term $\frac{5}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{2} lim_{x \to 0} \frac{1}{x \csc (5 x)}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{5}{2} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} x \csc (5 x)}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{5}{2} \cdot \frac{1}{lim_{x \to 0} x \csc (5 x)}$

Step 4

Consider the left sided limit.

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

Step 5

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

$\begin{array}{c} x & x \csc (5 x) \\ - 0.1 & 0.20858296 \\ - 0.01 & 0.20008335 \\ - 0.001 & 0.20000083 \end{array}$

Step 6

As the $x$ values approach $0$ , the function values approach $0.2$ . Thus, the limit of $x \csc (5 x)$ as $x$ approaches $0$ from the left is $0.2$ .

$0.2$

Step 7

Consider the right sided limit.

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

Step 8

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

$\begin{array}{c} x & x \csc (5 x) \\ 0.1 & 0.20858296 \\ 0.01 & 0.20008335 \\ 0.001 & 0.20000083 \end{array}$

Step 9

As the $x$ values approach $0$ , the function values approach $0.2$ . Thus, the limit of $x \csc (5 x)$ as $x$ approaches $0$ from the right is $0.2$ .

$\frac{5}{2} \cdot \frac{1}{0.2}$

Step 10

Cancel the common factor of $0.2$ .

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Step 10.1

Factor $0.2$ out of $5$ .

$\frac{0.2 (25)}{2} \cdot \frac{1}{0.2}$

Step 10.2

Cancel the common factor.

$\frac{0.2 \cdot 25}{2} \cdot \frac{1}{0.2}$

Step 10.3

Rewrite the expression.

$\frac{25}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{25}{2}$

Decimal Form:

$12.5$