Precalculus Examples

$lim_{x \to 5} \frac{x^{2} - 25}{x^{2} + x - 30}$

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.

$\frac{lim_{x \to 5} x^{2} - 25}{lim_{x \to 5} x^{2} + x - 30}$

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

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} x^{2} - lim_{x \to 5} 25}{lim_{x \to 5} x^{2} + x - 30}$

Step 1.1.2.1.2

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

$\frac{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 25}{lim_{x \to 5} x^{2} + x - 30}$

Step 1.1.2.1.3

Evaluate the limit of $25$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 1 \cdot 25}{lim_{x \to 5} x^{2} + x - 30}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{5^{2} - 1 \cdot 25}{lim_{x \to 5} x^{2} + x - 30}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Raise $5$ to the power of $2$ .

$\frac{25 - 1 \cdot 25}{lim_{x \to 5} x^{2} + x - 30}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $25$ .

$\frac{25 - 25}{lim_{x \to 5} x^{2} + x - 30}$

Step 1.1.2.3.2

Subtract $25$ from $25$ .

$\frac{0}{lim_{x \to 5} x^{2} + x - 30}$

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 $5$ .

$\frac{0}{lim_{x \to 5} x^{2} + lim_{x \to 5} x - lim_{x \to 5} 30}$

Step 1.1.3.2

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

$\frac{0}{{(lim_{x \to 5} x)}^{2} + lim_{x \to 5} x - lim_{x \to 5} 30}$

Step 1.1.3.3

Evaluate the limit of $30$ which is constant as $x$ approaches $5$ .

$\frac{0}{{(lim_{x \to 5} x)}^{2} + lim_{x \to 5} x - 1 \cdot 30}$

Step 1.1.3.4

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 1.1.3.4.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{0}{5^{2} + lim_{x \to 5} x - 1 \cdot 30}$

Step 1.1.3.4.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{0}{5^{2} + 5 - 1 \cdot 30}$

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

Raise $5$ to the power of $2$ .

$\frac{0}{25 + 5 - 1 \cdot 30}$

Step 1.1.3.5.1.2

Multiply $- 1$ by $30$ .

$\frac{0}{25 + 5 - 30}$

Step 1.1.3.5.2

Add $25$ and $5$ .

$\frac{0}{30 - 30}$

Step 1.1.3.5.3

Subtract $30$ from $30$ .

$\frac{0}{0}$

Step 1.1.3.5.4

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.

$lim_{x \to 5} \frac{x^{2} - 25}{x^{2} + x - 30} = lim_{x \to 5} \frac{\frac{d}{d x} [x^{2} - 25]}{\frac{d}{d x} [x^{2} + x - 30]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 5} \frac{\frac{d}{d x} [x^{2} - 25]}{\frac{d}{d x} [x^{2} + x - 30]}$

Step 1.3.2

By the Sum Rule, the derivative of $x^{2} - 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25]$ .

$lim_{x \to 5} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25]}{\frac{d}{d x} [x^{2} + x - 30]}$

Step 1.3.3

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

$lim_{x \to 5} \frac{2 x + \frac{d}{d x} [- 25]}{\frac{d}{d x} [x^{2} + x - 30]}$

Step 1.3.4

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{2 x + 0}{\frac{d}{d x} [x^{2} + x - 30]}$

Step 1.3.5

Add $2 x$ and $0$ .

$lim_{x \to 5} \frac{2 x}{\frac{d}{d x} [x^{2} + x - 30]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{2} + x - 30$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 30]$ .

$lim_{x \to 5} \frac{2 x}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 30]}$

Step 1.3.7

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

$lim_{x \to 5} \frac{2 x}{2 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 30]}$

Step 1.3.8

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

$lim_{x \to 5} \frac{2 x}{2 x + 1 + \frac{d}{d x} [- 30]}$

Step 1.3.9

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{2 x}{2 x + 1 + 0}$

Step 1.3.10

Add $2 x + 1$ and $0$ .

$lim_{x \to 5} \frac{2 x}{2 x + 1}$

Step 2

Evaluate the limit.

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Step 2.1

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

$2 lim_{x \to 5} \frac{x}{2 x + 1}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$2 \frac{lim_{x \to 5} x}{lim_{x \to 5} 2 x + 1}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$2 \frac{lim_{x \to 5} x}{lim_{x \to 5} 2 x + lim_{x \to 5} 1}$

Step 2.4

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

$2 \frac{lim_{x \to 5} x}{2 lim_{x \to 5} x + lim_{x \to 5} 1}$

Step 2.5

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$2 \frac{lim_{x \to 5} x}{2 lim_{x \to 5} x + 1}$

Step 3

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$2 \frac{5}{2 lim_{x \to 5} x + 1}$

Step 3.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$2 \frac{5}{2 \cdot 5 + 1}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $2$ by $5$ .

$2 \frac{5}{10 + 1}$

Step 4.1.2

Add $10$ and $1$ .

$2 (\frac{5}{11})$

Step 4.2

Multiply $2 (\frac{5}{11})$ .

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Step 4.2.1

Combine $2$ and $\frac{5}{11}$ .

$\frac{2 \cdot 5}{11}$

Step 4.2.2

Multiply $2$ by $5$ .

$\frac{10}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{10}{11}$

Decimal Form:

$0.\overline{90}$