Calculus Examples

$lim_{x \to 0} \frac{8}{x \cot (5 x)}$

Step 1

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$8 lim_{x \to 0} \frac{1}{x \cot (5 x)}$

Step 2

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

$8 \frac{lim_{x \to 0} 1}{lim_{x \to 0} x \cot (5 x)}$

Step 3

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

$8 \frac{1}{lim_{x \to 0} x \cot (5 x)}$

Step 4

Consider the left sided limit.

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

Step 5

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

$\begin{array}{c} x & x \cot (5 x) \\ - 0.1 & 0.18304877 \\ - 0.01 & 0.1998333 \\ - 0.001 & 0.19999833 \end{array}$

Step 6

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

$0.2$

Step 7

Consider the right sided limit.

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

Step 8

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

$\begin{array}{c} x & x \cot (5 x) \\ 0.1 & 0.18304877 \\ 0.01 & 0.1998333 \\ 0.001 & 0.19999833 \end{array}$

Step 9

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

$8 (\frac{1}{0.2})$

Step 10

Cancel the common factor of $0.2$ .

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Step 10.1

Factor $0.2$ out of $8$ .

$0.2 (40) \frac{1}{0.2}$

Step 10.2

Cancel the common factor.

$0.2 \cdot 40 \frac{1}{0.2}$

Step 10.3

Rewrite the expression.

$40$