Calculus Examples

$lim_{x \to 0} \frac{\sin (5 x)}{x}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} \sin (5 x)}{lim_{x \to 0} x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{\sin (lim_{x \to 0} 5 x)}{lim_{x \to 0} x}$

Step 1.1.2.1.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (5 lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 1.1.2.2

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

$\frac{\sin (5 \cdot 0)}{lim_{x \to 0} x}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $5$ by $0$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.3

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

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to 0} \frac{\sin (5 x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (5 x)]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (5 x)]}{\frac{d}{d x} [x]}$

Step 1.3.2

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 $g (x) = 5 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x]}{\frac{d}{d x} [x]}$

Step 1.3.2.2

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

$lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [5 x]}{\frac{d}{d x} [x]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to 0} \frac{\cos (5 x) \frac{d}{d x} [5 x]}{\frac{d}{d x} [x]}$

Step 1.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\cos (5 x) (5 \frac{d}{d x} [x])}{\frac{d}{d x} [x]}$

Step 1.3.4

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

$lim_{x \to 0} \frac{\cos (5 x) (5 \cdot 1)}{\frac{d}{d x} [x]}$

Step 1.3.5

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{\cos (5 x) \cdot 5}{\frac{d}{d x} [x]}$

Step 1.3.6

Move $5$ to the left of $\cos (5 x)$ .

$lim_{x \to 0} \frac{5 \cdot \cos (5 x)}{\frac{d}{d x} [x]}$

Step 1.3.7

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

$lim_{x \to 0} \frac{5 \cos (5 x)}{1}$

Step 1.4

Divide $5 \cos (5 x)$ by $1$ .

$lim_{x \to 0} 5 \cos (5 x)$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$5 lim_{x \to 0} \cos (5 x)$

Step 2.2

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

$5 \cos (lim_{x \to 0} 5 x)$

Step 2.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$5 \cos (5 lim_{x \to 0} x)$

Step 3

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

$5 \cos (5 \cdot 0)$

Step 4

Simplify the answer.

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Step 4.1

Multiply $5$ by $0$ .

$5 \cos (0)$

Step 4.2

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

$5 \cdot 1$

Step 4.3

Multiply $5$ by $1$ .

$5$