Calculus Examples

$lim_{y \to 0} \frac{\sin (3 y) \cot (5 y)}{y \cdot \cot (4 y)}$

Step 1

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

$\frac{lim_{y \to 0} \sin (3 y) \cot (5 y)}{lim_{y \to 0} y \cdot \cot (4 y)}$

Step 2

Consider the left sided limit.

$lim_{y \to 0^{-}} \sin (3 y) \cot (5 y)$

Step 3

Make a table to show the behavior of the function $\sin (3 y) \cot (5 y)$ as $y$ approaches $0$ from the left.

$\begin{array}{c} y & \sin (3 y) \cot (5 y) \\ - 0.1 & 0.5409461 \\ - 0.01 & 0.59940999 \\ - 0.001 & 0.5999941 \\ - 0.0001 & 0.59999994 \end{array}$

Step 4

As the $y$ values approach $0$ , the function values approach $0.6$ . Thus, the limit of $\sin (3 y) \cot (5 y)$ as $y$ approaches $0$ from the left is $0.6$ .

$0.6$

Step 5

Consider the right sided limit.

$lim_{y \to 0^{+}} \sin (3 y) \cot (5 y)$

Step 6

Make a table to show the behavior of the function $\sin (3 y) \cot (5 y)$ as $y$ approaches $0$ from the right.

$\begin{array}{c} y & \sin (3 y) \cot (5 y) \\ 0.1 & 0.5409461 \\ 0.01 & 0.59940999 \\ 0.001 & 0.5999941 \\ 0.0001 & 0.59999994 \end{array}$

Step 7

As the $y$ values approach $0$ , the function values approach $0.6$ . Thus, the limit of $\sin (3 y) \cot (5 y)$ as $y$ approaches $0$ from the right is $0.6$ .

$\frac{0.6}{lim_{y \to 0} y \cdot \cot (4 y)}$

Step 8

Consider the left sided limit.

$lim_{y \to 0^{-}} y \cdot \cot (4 y)$

Step 9

Make a table to show the behavior of the function $y \cdot \cot (4 y)$ as $y$ approaches $0$ from the left.

$\begin{array}{c} y & y \cdot \cot (4 y) \\ - 0.1 & 0.23652224 \\ - 0.01 & 0.24986665 \\ - 0.001 & 0.24999866 \end{array}$

Step 10

As the $y$ values approach $0$ , the function values approach $0.25$ . Thus, the limit of $y \cdot \cot (4 y)$ as $y$ approaches $0$ from the left is $0.25$ .

$0.25$

Step 11

Consider the right sided limit.

$lim_{y \to 0^{+}} y \cdot \cot (4 y)$

Step 12

Make a table to show the behavior of the function $y \cdot \cot (4 y)$ as $y$ approaches $0$ from the right.

$\begin{array}{c} y & y \cdot \cot (4 y) \\ 0.1 & 0.23652224 \\ 0.01 & 0.24986665 \\ 0.001 & 0.24999866 \end{array}$

Step 13

As the $y$ values approach $0$ , the function values approach $0.25$ . Thus, the limit of $y \cdot \cot (4 y)$ as $y$ approaches $0$ from the right is $0.25$ .

$\frac{0.6}{0.25}$

Step 14

Divide $0.6$ by $0.25$ .

$2.4$