Calculus Examples

$lim_{n \to (\frac{π}{2})} \frac{\cos (n)}{\sin (2 n)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\frac{\cos (n)}{\sin (2 n)}$ as a product.

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

Step 1.2

Write $\cos (n)$ as a fraction with denominator $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Divide $\cos (n)$ by $1$ .

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

Step 1.3.2

Convert from $\frac{1}{\sin (2 n)}$ to $\csc (2 n)$ .

$lim_{n \to (\frac{π}{2})} \cos (n) \csc (2 n)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $(\frac{π}{2})$ .

$lim_{n \to (\frac{π}{2})} \cos (n) \cdot lim_{n \to (\frac{π}{2})} \csc (2 n)$

Step 3

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

$\cos (lim_{n \to (\frac{π}{2})} n) \cdot lim_{n \to (\frac{π}{2})} \csc (2 n)$

Step 4

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

$\cos (lim_{n \to (\frac{π}{2})} n) \cdot \csc (lim_{n \to (\frac{π}{2})} 2 n)$

Step 5

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

$\cos (lim_{n \to (\frac{π}{2})} n) \cdot \csc (2 lim_{n \to (\frac{π}{2})} n)$

Step 6

Evaluate the limits by plugging in $(\frac{π}{2})$ for all occurrences of $n$ .

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Step 6.1

Evaluate the limit of $n$ by plugging in $(\frac{π}{2})$ for $n$ .

$\cos ((\frac{π}{2})) \cdot \csc (2 lim_{n \to (\frac{π}{2})} n)$

Step 6.2

Evaluate the limit of $n$ by plugging in $(\frac{π}{2})$ for $n$ .

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

Step 7

Simplify the answer.

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Step 7.1

Multiply $2$ by each element of the matrix.

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

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

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

Step 7.4

Remove parentheses.

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

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

Step 7.5

Simplify the denominator.

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Step 7.5.1

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

$\cos ((\frac{π}{2})) \cdot \frac{1}{\sin (0)}$

Step 7.5.2

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

$\cos ((\frac{π}{2})) \cdot \frac{1}{0}$

Step 7.5.3

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

Undefined

$\cos ((\frac{π}{2})) \cdot \frac{1}{0}$

Step 7.6

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

Undefined

$\cos ((\frac{π}{2})) \cdot \frac{1}{0}$

Step 8

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

Undefined