Calculus Examples

$lim_{y \to 0} 5 y^{2 (\cot (- 5 y)) (\csc (- 8 y))}$

Step 1

Set up the limit as a left-sided limit.

$lim_{y \to 0^{-}} 5 y^{2 (\cot (- 5 y)) (\csc (- 8 y))}$

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of $5 y^{2 (\cot (- 5 y)) (\csc (- 8 y))}$ by plugging in $0$ for $y$ .

$5 \cdot 0^{2 \cot (- 5 \cdot 0) \csc (- 8 \cdot 0)}$

Step 2.2

Rewrite $\cot (- 5 \cdot 0)$ in terms of sines and cosines.

$5 \cdot 0^{2 \frac{\cos (- 5 \cdot 0)}{\sin (- 5 \cdot 0)} \csc (- 8 \cdot 0)}$

Step 2.3

Multiply $- 5$ by $0$ .

$5 \cdot 0^{2 \frac{\cos (- 5 \cdot 0)}{\sin (0)} \csc (- 8 \cdot 0)}$

Step 2.4

The exact value of $\sin (0)$ is $0$ .

$5 \cdot 0^{2 \frac{\cos (- 5 \cdot 0)}{0} \csc (- 8 \cdot 0)}$

Step 2.5

Since $5 \cdot 0^{2 \frac{\cos (- 5 \cdot 0)}{0} \csc (- 8 \cdot 0)}$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{y \to 0^{+}} 5 y^{2 (\cot (- 5 y)) (\csc (- 8 y))}$

Step 4

Evaluate the right-sided limit.

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Step 4.1

Move the term $5$ outside of the limit because it is constant with respect to $y$ .

$5 lim_{y \to 0^{+}} y^{2 (\cot (- 5 y)) (\csc (- 8 y))}$

Step 4.2

Use the properties of logarithms to simplify the limit.

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Step 4.2.1

Rewrite $y^{2 (\cot (- 5 y)) (\csc (- 8 y))}$ as $e^{\ln (y^{2 (\cot (- 5 y)) (\csc (- 8 y))})}$ .

$5 lim_{y \to 0^{+}} e^{\ln (y^{2 (\cot (- 5 y)) (\csc (- 8 y))})}$

Step 4.2.2

Expand $\ln (y^{2 (\cot (- 5 y)) (\csc (- 8 y))})$ by moving $2 (\cot (- 5 y)) (\csc (- 8 y))$ outside the logarithm.

$5 lim_{y \to 0^{+}} e^{2 (\cot (- 5 y)) (\csc (- 8 y)) \ln (y)}$

Step 4.3

Since the exponent $2 (\cot (- 5 y)) (\csc (- 8 y)) \ln (y)$ approaches $- \infty$ , the quantity $e^{2 (\cot (- 5 y)) (\csc (- 8 y)) \ln (y)}$ approaches $0$ .

$5 \cdot 0$

Step 4.4

Multiply $5$ by $0$ .

$0$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$