Calculus Examples

$lim_{x \to \frac{1}{2} \cdot π} \frac{\cos (2 x)}{\tan (x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$lim_{x \to \frac{1}{2} \cdot π} \frac{\cos (2 x)}{\frac{\sin (x)}{\cos (x)}}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$lim_{x \to \frac{1}{2} \cdot π} \frac{\cos (2 x) \cos (x)}{\sin (x)}$

Step 1.3

Convert from $\frac{\cos (2 x) \cos (x)}{\sin (x)}$ to $\cos (2 x) \cot (x)$ .

$lim_{x \to \frac{1}{2} \cdot π} \cos (2 x) \cot (x)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{1}{2} \cdot π$ .

$lim_{x \to \frac{1}{2} \cdot π} \cos (2 x) \cdot lim_{x \to \frac{1}{2} \cdot π} \cot (x)$

Step 3

Move the limit inside the trig function because cosine is continuous.

$\cos (lim_{x \to \frac{1}{2} \cdot π} 2 x) \cdot lim_{x \to \frac{1}{2} \cdot π} \cot (x)$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\cos (2 lim_{x \to \frac{1}{2} \cdot π} x) \cdot lim_{x \to \frac{1}{2} \cdot π} \cot (x)$

Step 5

Move the limit inside the trig function because cotangent is continuous.

$\cos (2 lim_{x \to \frac{1}{2} \cdot π} x) \cdot \cot (lim_{x \to \frac{1}{2} \cdot π} x)$

Step 6

Evaluate the limits by plugging in $\frac{1}{2} \cdot π$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $\frac{1}{2} \cdot π$ for $x$ .

$\cos (2 (\frac{1}{2} \cdot π)) \cdot \cot (lim_{x \to \frac{1}{2} \cdot π} x)$

Step 6.2

Combine $\frac{1}{2}$ and $π$ .

$\cos (2 \frac{π}{2}) \cdot \cot (lim_{x \to \frac{1}{2} \cdot π} x)$

Step 6.3

Evaluate the limit of $x$ by plugging in $\frac{1}{2} \cdot π$ for $x$ .

$\cos (2 \frac{π}{2}) \cdot \cot (\frac{1}{2} \cdot π)$

Step 6.4

Combine $\frac{1}{2}$ and $π$ .

$\cos (2 \frac{π}{2}) \cdot \cot (\frac{π}{2})$

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Cancel the common factor.

$\cos (2 \frac{π}{2}) \cdot \cot (\frac{π}{2})$

Step 7.1.2

Rewrite the expression.

$\cos (π) \cdot \cot (\frac{π}{2})$

Step 7.2

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.

$- \cos (0) \cdot \cot (\frac{π}{2})$

Step 7.3

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

$- 1 \cdot 1 \cdot \cot (\frac{π}{2})$

Step 7.4

Multiply $- 1$ by $1$ .

$- 1 \cdot \cot (\frac{π}{2})$

Step 7.5

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$- 1 \cdot 0$

Step 7.6

Multiply $- 1$ by $0$ .

$0$