Calculus Examples

$lim_{x \to (\frac{π}{4})} {(1 + \sin (4 x))}^{\cot (4 x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + \sin (4 x))}^{\cot (4 x)}$ as $e^{\ln ({(1 + \sin (4 x))}^{\cot (4 x)})}$ .

$lim_{x \to (\frac{π}{4})} e^{\ln ({(1 + \sin (4 x))}^{\cot (4 x)})}$

Step 1.2

Expand $\ln ({(1 + \sin (4 x))}^{\cot (4 x)})$ by moving $\cot (4 x)$ outside the logarithm.

$lim_{x \to (\frac{π}{4})} e^{\cot (4 x) \ln (1 + \sin (4 x))}$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to {(\frac{π}{4})}^{-}} e^{\cot (4 x) \ln (1 + \sin (4 x))}$

Step 3

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

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Step 3.1

Evaluate the limit of $e^{\cot (4 x) \ln (1 + \sin (4 x))}$ by plugging in $(\frac{π}{4})$ for $x$ .

$e^{\cot (4 (\frac{π}{4})) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.2

Multiply $4$ by each element of the matrix.

$e^{\cot ((4 \frac{π}{4})) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Cancel the common factor.

$e^{\cot ((4 \frac{π}{4})) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.3.2

Rewrite the expression.

$e^{\cot ((π)) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.4

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

$e^{\frac{\cos ((π))}{\sin ((π))} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.5

Remove parentheses.

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Step 3.5.1

Cancel the common factor.

$e^{\frac{\cos (4 \frac{π}{4})}{\sin ((π))} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.5.2

Rewrite the expression.

$e^{\frac{\cos (π)}{\sin ((π))} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.6

Remove parentheses.

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Step 3.6.1

Cancel the common factor.

$e^{\frac{\cos (π)}{\sin (4 \frac{π}{4})} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.6.2

Rewrite the expression.

$e^{\frac{\cos (π)}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$e^{\frac{- \cos (0)}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.7.2

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{- 1 \cdot 1}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.7.3

Multiply $- 1$ by $1$ .

$e^{\frac{- 1}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.8

Simplify the denominator.

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Step 3.8.1

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

$e^{\frac{- 1}{\sin (0)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.8.2

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

$e^{\frac{- 1}{0} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.8.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{- 1}{0} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 3.9

Since $e^{\frac{- 1}{0} \ln (1 + \sin (4 (\frac{π}{4})))}$ is undefined, the limit does not exist.

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to {(\frac{π}{4})}^{+}} e^{\cot (4 x) \ln (1 + \sin (4 x))}$

Step 5

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

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Step 5.1

Evaluate the limit of $e^{\cot (4 x) \ln (1 + \sin (4 x))}$ by plugging in $(\frac{π}{4})$ for $x$ .

$e^{\cot (4 (\frac{π}{4})) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.2

Multiply $4$ by each element of the matrix.

$e^{\cot ((4 \frac{π}{4})) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.3

Cancel the common factor of $4$ .

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Step 5.3.1

Cancel the common factor.

$e^{\cot ((4 \frac{π}{4})) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.3.2

Rewrite the expression.

$e^{\cot ((π)) \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.4

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

$e^{\frac{\cos ((π))}{\sin ((π))} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.5

Remove parentheses.

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Step 5.5.1

Cancel the common factor.

$e^{\frac{\cos (4 \frac{π}{4})}{\sin ((π))} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.5.2

Rewrite the expression.

$e^{\frac{\cos (π)}{\sin ((π))} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.6

Remove parentheses.

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Step 5.6.1

Cancel the common factor.

$e^{\frac{\cos (π)}{\sin (4 \frac{π}{4})} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.6.2

Rewrite the expression.

$e^{\frac{\cos (π)}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$e^{\frac{- \cos (0)}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.7.2

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{- 1 \cdot 1}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.7.3

Multiply $- 1$ by $1$ .

$e^{\frac{- 1}{\sin (π)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.8

Simplify the denominator.

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Step 5.8.1

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

$e^{\frac{- 1}{\sin (0)} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.8.2

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

$e^{\frac{- 1}{0} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.8.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{- 1}{0} \ln (1 + \sin (4 (\frac{π}{4})))}$

Step 5.9

Since $e^{\frac{- 1}{0} \ln (1 + \sin (4 (\frac{π}{4})))}$ 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.

$Does not exist$