Calculus Examples

$lim_{x \to π} {\sin (x)}^{\csc (x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${\sin (x)}^{\csc (x)}$ as $e^{\ln ({\sin (x)}^{\csc (x)})}$ .

$lim_{x \to π} e^{\ln ({\sin (x)}^{\csc (x)})}$

Step 1.2

Expand $\ln ({\sin (x)}^{\csc (x)})$ by moving $\csc (x)$ outside the logarithm.

$lim_{x \to π} e^{\csc (x) \ln (\sin (x))}$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to π^{-}} e^{\csc (x) \ln (\sin (x))}$

Step 3

Since the exponent $\csc (x) \ln (\sin (x))$ approaches $- \infty$ , the quantity $e^{\csc (x) \ln (\sin (x))}$ approaches $0$ .

$0$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to π^{+}} e^{\csc (x) \ln (\sin (x))}$

Step 5

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

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Step 5.1

Evaluate the limit of $e^{\csc (x) \ln (\sin (x))}$ by plugging in $π$ for $x$ .

$e^{\csc (π) \ln (\sin (π))}$

Step 5.2

Rewrite $\csc (π)$ in terms of sines and cosines.

$e^{\frac{1}{\sin (π)} \ln (\sin (π))}$

Step 5.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$e^{\frac{1}{\sin (0)} \ln (\sin (π))}$

Step 5.4

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

$e^{\frac{1}{0} \ln (\sin (π))}$

Step 5.5

Since $e^{\frac{1}{0} \ln (\sin (π))}$ is undefined, the limit does not exist.

$Does not exist$

Step 6

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

$Does not exist$