Calculus Examples

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

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.

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

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 exponent $2$ from ${(x + 5)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Add $- 5$ and $5$ .

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

Step 1.1.2.3.2

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

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

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 $- 5$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

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 $x$ .

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

Step 1.1.3.4.2

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

$\frac{0}{{(- 5)}^{2} + 5 \cdot - 5}$

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 $2$ .

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

Step 1.1.3.5.1.2

Multiply $5$ by $- 5$ .

$\frac{0}{25 - 25}$

Step 1.1.3.5.2

Subtract $25$ from $25$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

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

Step 1.3.3

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.3.3.1

Apply the distributive property.

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

Step 1.3.3.2

Apply the distributive property.

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

Step 1.3.3.3

Apply the distributive property.

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

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $x$ by $x$ .

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

Step 1.3.4.1.2

Move $5$ to the left of $x$ .

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

Step 1.3.4.1.3

Multiply $5$ by $5$ .

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

Step 1.3.4.2

Add $5 x$ and $5 x$ .

$lim_{x \to {(- 5)}^{+}} \frac{\frac{d}{d x} [x^{2} + 10 x + 25]}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.5

By the Sum Rule, the derivative of $x^{2} + 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$ .

$lim_{x \to {(- 5)}^{+}} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.6

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

$lim_{x \to {(- 5)}^{+}} \frac{2 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.7

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.8

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

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10 \cdot 1 + \frac{d}{d x} [25]}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.9

Multiply $10$ by $1$ .

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10 + \frac{d}{d x} [25]}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.10

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

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10 + 0}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.11

Add $2 x + 10$ and $0$ .

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10}{\frac{d}{d x} [x^{2} + 5 x]}$

Step 1.3.12

By the Sum Rule, the derivative of $x^{2} + 5 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x]$ .

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x]}$

Step 1.3.13

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

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10}{2 x + \frac{d}{d x} [5 x]}$

Step 1.3.14

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.3.14.1

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

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10}{2 x + 5 \frac{d}{d x} [x]}$

Step 1.3.14.2

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

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10}{2 x + 5 \cdot 1}$

Step 1.3.14.3

Multiply $5$ by $1$ .

$lim_{x \to {(- 5)}^{+}} \frac{2 x + 10}{2 x + 5}$

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 $- 5$ .

$\frac{lim_{x \to {(- 5)}^{+}} 2 x + 10}{lim_{x \to {(- 5)}^{+}} 2 x + 5}$

Step 2.2

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

$\frac{lim_{x \to {(- 5)}^{+}} 2 x + lim_{x \to {(- 5)}^{+}} 10}{lim_{x \to {(- 5)}^{+}} 2 x + 5}$

Step 2.3

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

$\frac{2 lim_{x \to {(- 5)}^{+}} x + lim_{x \to {(- 5)}^{+}} 10}{lim_{x \to {(- 5)}^{+}} 2 x + 5}$

Step 2.4

Evaluate the limit of $10$ which is constant as $x$ approaches $- 5$ .

$\frac{2 lim_{x \to {(- 5)}^{+}} x + 10}{lim_{x \to {(- 5)}^{+}} 2 x + 5}$

Step 2.5

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

$\frac{2 lim_{x \to {(- 5)}^{+}} x + 10}{lim_{x \to {(- 5)}^{+}} 2 x + lim_{x \to {(- 5)}^{+}} 5}$

Step 2.6

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

$\frac{2 lim_{x \to {(- 5)}^{+}} x + 10}{2 lim_{x \to {(- 5)}^{+}} x + lim_{x \to {(- 5)}^{+}} 5}$

Step 2.7

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

$\frac{2 lim_{x \to {(- 5)}^{+}} x + 10}{2 lim_{x \to {(- 5)}^{+}} x + 5}$

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 $x$ .

$\frac{2 \cdot - 5 + 10}{2 lim_{x \to {(- 5)}^{+}} x + 5}$

Step 3.2

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

$\frac{2 \cdot - 5 + 10}{2 \cdot - 5 + 5}$

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $2 \cdot - 5 + 10$ and $2 \cdot - 5 + 5$ .

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Step 4.1.1

Factor $5$ out of $2 \cdot - 5$ .

$\frac{5 (2 \cdot - 1) + 10}{2 \cdot - 5 + 5}$

Step 4.1.2

Factor $5$ out of $10$ .

$\frac{5 (2 \cdot - 1) + 5 \cdot 2}{2 \cdot - 5 + 5}$

Step 4.1.3

Factor $5$ out of $5 (2 \cdot - 1) + 5 (2)$ .

$\frac{5 (2 \cdot - 1 + 2)}{2 \cdot - 5 + 5}$

Step 4.1.4

Cancel the common factors.

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Step 4.1.4.1

Factor $5$ out of $2 \cdot - 5$ .

$\frac{5 (2 \cdot - 1 + 2)}{5 (2 \cdot - 1) + 5}$

Step 4.1.4.2

Factor $5$ out of $5$ .

$\frac{5 (2 \cdot - 1 + 2)}{5 (2 \cdot - 1) + 5 (1)}$

Step 4.1.4.3

Factor $5$ out of $5 (2 \cdot - 1) + 5 (1)$ .

$\frac{5 (2 \cdot - 1 + 2)}{5 (2 \cdot - 1 + 1)}$

Step 4.1.4.4

Cancel the common factor.

$\frac{5 (2 \cdot - 1 + 2)}{5 (2 \cdot - 1 + 1)}$

Step 4.1.4.5

Rewrite the expression.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Multiply $2$ by $- 1$ .

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

Step 4.2.2

Add $- 2$ and $2$ .

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

Multiply $2$ by $- 1$ .

$\frac{0}{- 2 + 1}$

Step 4.3.2

Add $- 2$ and $1$ .

$\frac{0}{- 1}$

Step 4.4

Divide $0$ by $- 1$ .

$0$