Calculus Examples

$lim_{x \to 0} \frac{\sin (x)}{x^{2} + 3 x}$

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.

$\frac{lim_{x \to 0} \sin (x)}{lim_{x \to 0} x^{2} + 3 x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{\sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{2} + 3 x}$

Step 1.1.2.2

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

$\frac{\sin (0)}{lim_{x \to 0} x^{2} + 3 x}$

Step 1.1.2.3

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

$\frac{0}{lim_{x \to 0} x^{2} + 3 x}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{lim_{x \to 0} x^{2} + lim_{x \to 0} 3 x}$

Step 1.1.3.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 3 x}$

Step 1.1.3.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x}$

Step 1.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.1.3.4.1

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

$\frac{0}{0^{2} + 3 lim_{x \to 0} x}$

Step 1.1.3.4.2

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

$\frac{0}{0^{2} + 3 \cdot 0}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 + 3 \cdot 0}$

Step 1.1.3.5.1.2

Multiply $3$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sin (x)}{x^{2} + 3 x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2} + 3 x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2} + 3 x]}$

Step 1.3.2

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

$lim_{x \to 0} \frac{\cos (x)}{\frac{d}{d x} [x^{2} + 3 x]}$

Step 1.3.3

By the Sum Rule, the derivative of $x^{2} + 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x]$ .

$lim_{x \to 0} \frac{\cos (x)}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x]}$

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{\cos (x)}{2 x + \frac{d}{d x} [3 x]}$

Step 1.3.5

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.3.5.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\cos (x)}{2 x + 3 \frac{d}{d x} [x]}$

Step 1.3.5.2

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

$lim_{x \to 0} \frac{\cos (x)}{2 x + 3 \cdot 1}$

Step 1.3.5.3

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{\cos (x)}{2 x + 3}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \cos (x)}{lim_{x \to 0} 2 x + 3}$

Step 2.2

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

$\frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} 2 x + 3}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} 2 x + lim_{x \to 0} 3}$

Step 2.4

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

$\frac{\cos (lim_{x \to 0} x)}{2 lim_{x \to 0} x + lim_{x \to 0} 3}$

Step 2.5

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{\cos (lim_{x \to 0} x)}{2 lim_{x \to 0} x + 3}$

Step 3

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1

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

$\frac{\cos (0)}{2 lim_{x \to 0} x + 3}$

Step 3.2

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

$\frac{\cos (0)}{2 \cdot 0 + 3}$

Step 4

Simplify the answer.

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Step 4.1

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

$\frac{1}{2 \cdot 0 + 3}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $0$ .

$\frac{1}{0 + 3}$

Step 4.2.2

Add $0$ and $3$ .

$\frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$