Evaluate the Limit limit as n approaches (pi/2) of (cos(n))/(sin(2n))

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

Apply trigonometric identities.

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

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

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

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

Simplify.

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

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

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

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

Take the limit of each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.50000000 \dots$

Evaluate the Limit limit as n approaches (pi/2) of (cos(n))/(sin(2n))

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