Calculus Examples

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

Step 1

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

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

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.

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

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.

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

Step 2.1.2.1.2

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $\frac{1}{2}$ by $0$ .

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

Step 2.1.2.3.2

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

$\frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

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 $g (x) = \frac{x}{2}$ .

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 2.3.4

Combine $\frac{1}{2}$ and $\cos (\frac{x}{2})$ .

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

Step 2.3.5

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

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

Step 2.3.6

Multiply $\frac{\cos (\frac{x}{2})}{2}$ by $1$ .

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

Step 2.3.7

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

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

Multiply $\frac{\cos (\frac{x}{2})}{2}$ by $1$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Multiply $\frac{1}{2}$ by $0$ .

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

Step 5.2

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

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

Step 5.3

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

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$