Calculus Examples

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

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Cancel the common factor of $2$ .

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Step 1.2.3.1.1

Cancel the common factor.

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

Step 1.2.3.1.2

Rewrite the expression.

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

Step 1.2.3.2

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

$\frac{\sin (0)}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.2.3.3

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

$\frac{0}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{\cos (lim_{x \to \frac{π}{2}} x)}$

Step 1.3.2

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

$\frac{0}{\cos (\frac{π}{2})}$

Step 1.3.3

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

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{π}{2}} \frac{\sin (2 x)}{\cos (x)} = lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\sin (2 x)]}{\frac{d}{d x} [\cos (x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\sin (2 x)]}{\frac{d}{d x} [\cos (x)]}$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]}{\frac{d}{d x} [\cos (x)]}$

Step 3.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$lim_{x \to \frac{π}{2}} \frac{\cos (u) \frac{d}{d x} [2 x]}{\frac{d}{d x} [\cos (x)]}$

Step 3.2.3

Replace all occurrences of $u$ with $2 x$ .

$lim_{x \to \frac{π}{2}} \frac{\cos (2 x) \frac{d}{d x} [2 x]}{\frac{d}{d x} [\cos (x)]}$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to \frac{π}{2}} \frac{\cos (2 x) (2 \frac{d}{d x} [x])}{\frac{d}{d x} [\cos (x)]}$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \frac{π}{2}} \frac{\cos (2 x) (2 \cdot 1)}{\frac{d}{d x} [\cos (x)]}$

Step 3.5

Multiply $2$ by $1$ .

$lim_{x \to \frac{π}{2}} \frac{\cos (2 x) \cdot 2}{\frac{d}{d x} [\cos (x)]}$

Step 3.6

Move $2$ to the left of $\cos (2 x)$ .

$lim_{x \to \frac{π}{2}} \frac{2 \cdot \cos (2 x)}{\frac{d}{d x} [\cos (x)]}$

Step 3.7

Multiply $2$ by $\cos (2 x)$ .

$lim_{x \to \frac{π}{2}} \frac{2 \cos (2 x)}{\frac{d}{d x} [\cos (x)]}$

Step 3.8

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 4

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

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

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{2}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

$2 \frac{\cos (2 \frac{π}{2})}{- \sin (\frac{π}{2})}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Cancel the common factor of $2$ .

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Step 11.1.1.1

Cancel the common factor.

$2 \frac{\cos (2 \frac{π}{2})}{- \sin (\frac{π}{2})}$

Step 11.1.1.2

Rewrite the expression.

$2 \frac{\cos (π)}{- \sin (\frac{π}{2})}$

Step 11.1.2

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.

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

Step 11.1.3

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

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

Step 11.1.4

Multiply $- 1$ by $1$ .

$2 \frac{- 1}{- \sin (\frac{π}{2})}$

Step 11.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$2 \frac{- 1}{- 1 \cdot 1}$

Step 11.3

Multiply $- 1$ by $1$ .

$2 (\frac{- 1}{- 1})$

Step 11.4

Dividing two negative values results in a positive value.

$2 (\frac{1}{1})$

Step 11.5

Cancel the common factor of $1$ .

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Step 11.5.1

Cancel the common factor.

$2 (\frac{1}{1})$

Step 11.5.2

Rewrite the expression.

$2 \cdot 1$

Step 11.6

Multiply $2$ by $1$ .

$2$