Calculus Examples

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

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{\sin (- \frac{π x}{2})}{x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} \sin (- \frac{π x}{2})}{lim_{x \to 0} x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

$\frac{1}{2} \cdot \frac{\sin (lim_{x \to 0} - \frac{π x}{2})}{lim_{x \to 0} x}$

Step 2.1.2.1.2

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

$\frac{1}{2} \cdot \frac{\sin (- lim_{x \to 0} \frac{π x}{2})}{lim_{x \to 0} x}$

Step 2.1.2.1.3

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{\sin (- \frac{π}{2} lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{\sin (- \frac{π}{2} \cdot 0)}{lim_{x \to 0} x}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $\frac{π}{2}$ by $0$ .

$\frac{1}{2} \cdot \frac{\sin (- 0)}{lim_{x \to 0} x}$

Step 2.1.2.3.2

Multiply $- 1$ by $0$ .

$\frac{1}{2} \cdot \frac{\sin (0)}{lim_{x \to 0} x}$

Step 2.1.2.3.3

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

$\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.2

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.

$lim_{x \to 0} \frac{\sin (- \frac{π x}{2})}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (- \frac{π x}{2})]}{\frac{d}{d x} [x]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [\sin (- \frac{π x}{2})]}{\frac{d}{d x} [x]}$

Step 2.3.2

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 $g (x) = - \frac{π x}{2}$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $- \frac{π x}{2}$ .

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [- \frac{π x}{2}]}{\frac{d}{d x} [x]}$

Step 2.3.2.2

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

$\frac{1}{2} lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [- \frac{π x}{2}]}{\frac{d}{d x} [x]}$

Step 2.3.2.3

Replace all occurrences of $u$ with $- \frac{π x}{2}$ .

$\frac{1}{2} lim_{x \to 0} \frac{\cos (- \frac{π x}{2}) \frac{d}{d x} [- \frac{π x}{2}]}{\frac{d}{d x} [x]}$

Step 2.3.3

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

$\frac{1}{2} lim_{x \to 0} \frac{\cos (- \frac{π x}{2}) \cdot - 1 \frac{π}{2} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.4

Combine $\cos (- \frac{π x}{2})$ and $\frac{π}{2}$ .

$\frac{1}{2} lim_{x \to 0} \frac{- \frac{\cos (- \frac{π x}{2}) π}{2} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.5

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

$\frac{1}{2} lim_{x \to 0} \frac{- \frac{\cos (- \frac{π x}{2}) π}{2} \cdot 1}{\frac{d}{d x} [x]}$

Step 2.3.6

Multiply $- 1$ by $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{- \frac{\cos (- \frac{π x}{2}) π}{2}}{\frac{d}{d x} [x]}$

Step 2.3.7

Simplify the numerator.

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Step 2.3.7.1

Since $\cos (- \frac{π x}{2})$ is an even function, rewrite $\cos (- \frac{π x}{2})$ as $\cos (\frac{π x}{2})$ .

$\frac{1}{2} lim_{x \to 0} \frac{- \frac{\cos (\frac{π x}{2}) π}{2}}{\frac{d}{d x} [x]}$

Step 2.3.7.2

Reorder factors in $\cos (\frac{π x}{2}) π$ .

$\frac{1}{2} lim_{x \to 0} \frac{- \frac{π \cos (\frac{π x}{2})}{2}}{\frac{d}{d x} [x]}$

Step 2.3.8

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

$\frac{1}{2} lim_{x \to 0} \frac{- \frac{π \cos (\frac{π x}{2})}{2}}{1}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} lim_{x \to 0} - \frac{π \cos (\frac{π x}{2})}{2} \cdot 1$

Step 2.5

Multiply $- 1$ by $1$ .

$\frac{1}{2} lim_{x \to 0} - \frac{π \cos (\frac{π x}{2})}{2}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{1}{2} \cdot - 1 lim_{x \to 0} \frac{π \cos (\frac{π x}{2})}{2}$

Step 3.2

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot - 1 \frac{π}{2} lim_{x \to 0} \cos (\frac{π x}{2})$

Step 3.3

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

$\frac{1}{2} \cdot - 1 \frac{π}{2} \cos (lim_{x \to 0} \frac{π x}{2})$

Step 3.4

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot - 1 \frac{π}{2} \cos (\frac{π}{2} lim_{x \to 0} x)$

Step 4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 5.2

Move the negative in front of the fraction.

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

Step 5.3

Multiply $- \frac{1}{2} \cdot \frac{π}{2}$ .

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Step 5.3.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

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

Step 5.3.2

Multiply $2$ by $2$ .

$- \frac{π}{4} \cos (\frac{π}{2} \cdot 0)$

Step 5.4

Multiply $\frac{π}{2}$ by $0$ .

$- \frac{π}{4} \cos (0)$

Step 5.5

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

$- \frac{π}{4} \cdot 1$

Step 5.6

Multiply $- 1$ by $1$ .

$- \frac{π}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{4}$

Decimal Form:

$- 0.78539816 \dots$