Calculus Examples

$lim_{x \to 0} \frac{10 x \cos (x)}{\sin (x)}$

Step 1

Convert from $\frac{10 x \cos (x)}{\sin (x)}$ to $10 x \cot (x)$ .

$lim_{x \to 0} 10 x \cot (x)$

Step 2

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$10 lim_{x \to 0} x \cot (x)$

Step 3

Consider the left sided limit.

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

Step 4

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

$\begin{array}{c} x & x \cot (x) \\ - 0.1 & 0.99666444 \\ - 0.01 & 0.99996666 \\ - 0.001 & 0.99999966 \end{array}$

Step 5

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $x \cot (x)$ as $x$ approaches $0$ from the left is $1$ .

$1$

Step 6

Consider the right sided limit.

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

Step 7

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

$\begin{array}{c} x & x \cot (x) \\ 0.1 & 0.99666444 \\ 0.01 & 0.99996666 \\ 0.001 & 0.99999966 \end{array}$

Step 8

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $x \cot (x)$ as $x$ approaches $0$ from the right is $1$ .

$10 \cdot 1$

Step 9

Multiply $10$ by $1$ .

$10$