Calculus Examples

$lim_{x \to 0} (x) \cos (\frac{1}{x^{2}})$

Step 1

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

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

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 3

Consider the left sided limit.

$lim_{x \to 0^{-}} \cos (\frac{1}{x^{2}})$

Step 4

Make a table to show the behavior of the function $\cos (\frac{1}{x^{2}})$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \cos (\frac{1}{x^{2}}) \\ - 0.1 & 0.86231887 \\ - 0.01 & - 0.92173957 \\ - 0.001 & - 0.92173957 \end{array}$

Step 5

As the $x$ values approach $0$ , the function values approach $- 0.922$ . Thus, the limit of $\cos (\frac{1}{x^{2}})$ as $x$ approaches $0$ from the left is $- 0.922$ .

$- 0.922$

Step 6

Consider the right sided limit.

$lim_{x \to 0^{+}} \cos (\frac{1}{x^{2}})$

Step 7

Make a table to show the behavior of the function $\cos (\frac{1}{x^{2}})$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \cos (\frac{1}{x^{2}}) \\ 0.1 & 0.86231887 \\ 0.01 & - 0.92173957 \\ 0.001 & - 0.92173957 \end{array}$

Step 8

As the $x$ values approach $0$ , the function values approach $- 0.922$ . Thus, the limit of $\cos (\frac{1}{x^{2}})$ as $x$ approaches $0$ from the right is $- 0.922$ .

$0 \cdot - 0.922$

Step 9

Multiply $0$ by $- 0.922$ .

$0$