Calculus Examples

$lim_{x \to 0} 3 x \csc (2 x)$

Step 1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 0} x \csc (2 x)$

Step 2

Consider the left sided limit.

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

Step 3

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

$\begin{array}{c} x & x \csc (2 x) \\ - 0.1 & 0.50334895 \\ - 0.01 & 0.50003333 \\ - 0.001 & 0.50000033 \end{array}$

Step 4

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

$0.5$

Step 5

Consider the right sided limit.

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

Step 6

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

$\begin{array}{c} x & x \csc (2 x) \\ 0.1 & 0.50334895 \\ 0.01 & 0.50003333 \\ 0.001 & 0.50000033 \end{array}$

Step 7

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

$3 \cdot 0.5$

Step 8

Multiply $3$ by $0.5$ .

$1.5$