Calculus Examples

$lim_{x \to - 10} \frac{x^{2} + 10 x}{(x^{2} + 100) (x + 10)}$

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.

$\frac{lim_{x \to - 10} x^{2} + 10 x}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{lim_{x \to - 10} x^{2} + lim_{x \to - 10} 10 x}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to - 10} x)}^{2} + lim_{x \to - 10} 10 x}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.3

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to - 10} x)}^{2} + 10 lim_{x \to - 10} x}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.4

Evaluate the limits by plugging in $- 10$ for all occurrences of $x$ .

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Step 1.1.2.4.1

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{{(- 10)}^{2} + 10 lim_{x \to - 10} x}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.4.2

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{{(- 10)}^{2} + 10 \cdot - 10}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

Raise $- 10$ to the power of $2$ .

$\frac{100 + 10 \cdot - 10}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.5.1.2

Multiply $10$ by $- 10$ .

$\frac{100 - 100}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.2.5.2

Subtract $100$ from $100$ .

$\frac{0}{lim_{x \to - 10} (x^{2} + 100) (x + 10)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{0}{lim_{x \to - 10} x^{2} + 100 \cdot lim_{x \to - 10} x + 10}$

Step 1.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{0}{(lim_{x \to - 10} x^{2} + lim_{x \to - 10} 100) \cdot lim_{x \to - 10} x + 10}$

Step 1.1.3.3

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

$\frac{0}{({(lim_{x \to - 10} x)}^{2} + lim_{x \to - 10} 100) \cdot lim_{x \to - 10} x + 10}$

Step 1.1.3.4

Evaluate the limit of $100$ which is constant as $x$ approaches $- 10$ .

$\frac{0}{({(lim_{x \to - 10} x)}^{2} + 100) \cdot lim_{x \to - 10} x + 10}$

Step 1.1.3.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{0}{({(lim_{x \to - 10} x)}^{2} + 100) \cdot (lim_{x \to - 10} x + lim_{x \to - 10} 10)}$

Step 1.1.3.6

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

$\frac{0}{({(lim_{x \to - 10} x)}^{2} + 100) \cdot (lim_{x \to - 10} x + 10)}$

Step 1.1.3.7

Evaluate the limits by plugging in $- 10$ for all occurrences of $x$ .

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Step 1.1.3.7.1

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{0}{({(- 10)}^{2} + 100) \cdot (lim_{x \to - 10} x + 10)}$

Step 1.1.3.7.2

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{0}{({(- 10)}^{2} + 100) \cdot (- 10 + 10)}$

Step 1.1.3.8

Simplify the answer.

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Step 1.1.3.8.1

Raise $- 10$ to the power of $2$ .

$\frac{0}{(100 + 100) \cdot (- 10 + 10)}$

Step 1.1.3.8.2

Add $100$ and $100$ .

$\frac{0}{200 \cdot (- 10 + 10)}$

Step 1.1.3.8.3

Add $- 10$ and $10$ .

$\frac{0}{200 \cdot 0}$

Step 1.1.3.8.4

Multiply $200$ by $0$ .

$\frac{0}{0}$

Step 1.1.3.8.5

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

Undefined

$\frac{0}{0}$

Step 1.1.3.9

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.

$lim_{x \to - 10} \frac{x^{2} + 10 x}{(x^{2} + 100) (x + 10)} = lim_{x \to - 10} \frac{\frac{d}{d x} [x^{2} + 10 x]}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 10} \frac{\frac{d}{d x} [x^{2} + 10 x]}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3.2

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

$lim_{x \to - 10} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x]}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3.3

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

$lim_{x \to - 10} \frac{2 x + \frac{d}{d x} [10 x]}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 1.3.4.1

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

$lim_{x \to - 10} \frac{2 x + 10 \frac{d}{d x} [x]}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3.4.2

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

$lim_{x \to - 10} \frac{2 x + 10 \cdot 1}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3.4.3

Multiply $10$ by $1$ .

$lim_{x \to - 10} \frac{2 x + 10}{\frac{d}{d x} [(x^{2} + 100) (x + 10)]}$

Step 1.3.5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 100$ and $g (x) = x + 10$ .

$lim_{x \to - 10} \frac{2 x + 10}{(x^{2} + 100) \frac{d}{d x} [x + 10] + (x + 10) \frac{d}{d x} [x^{2} + 100]}$

Step 1.3.6

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

$lim_{x \to - 10} \frac{2 x + 10}{(x^{2} + 100) (\frac{d}{d x} [x] + \frac{d}{d x} [10]) + (x + 10) \frac{d}{d x} [x^{2} + 100]}$

Step 1.3.7

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

$lim_{x \to - 10} \frac{2 x + 10}{(x^{2} + 100) (1 + \frac{d}{d x} [10]) + (x + 10) \frac{d}{d x} [x^{2} + 100]}$

Step 1.3.8

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$lim_{x \to - 10} \frac{2 x + 10}{(x^{2} + 100) (1 + 0) + (x + 10) \frac{d}{d x} [x^{2} + 100]}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to - 10} \frac{2 x + 10}{(x^{2} + 100) \cdot 1 + (x + 10) \frac{d}{d x} [x^{2} + 100]}$

Step 1.3.10

Multiply $x^{2} + 100$ by $1$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + (x + 10) \frac{d}{d x} [x^{2} + 100]}$

Step 1.3.11

By the Sum Rule, the derivative of $x^{2} + 100$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [100]$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + (x + 10) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [100])}$

Step 1.3.12

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

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + (x + 10) (2 x + \frac{d}{d x} [100])}$

Step 1.3.13

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + (x + 10) (2 x + 0)}$

Step 1.3.14

Add $2 x$ and $0$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + (x + 10) (2 x)}$

Step 1.3.15

Move $2$ to the left of $x + 10$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 \cdot (x + 10) x}$

Step 1.3.16

Simplify.

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Step 1.3.16.1

Apply the distributive property.

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + (2 x + 2 \cdot 10) x}$

Step 1.3.16.2

Apply the distributive property.

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 x \cdot x + 2 \cdot 10 x}$

Step 1.3.16.3

Combine terms.

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Step 1.3.16.3.1

Raise $x$ to the power of $1$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 (x^{1} x) + 2 \cdot 10 x}$

Step 1.3.16.3.2

Raise $x$ to the power of $1$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 (x^{1} x^{1}) + 2 \cdot 10 x}$

Step 1.3.16.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 x^{1 + 1} + 2 \cdot 10 x}$

Step 1.3.16.3.4

Add $1$ and $1$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 x^{2} + 2 \cdot 10 x}$

Step 1.3.16.3.5

Multiply $2$ by $10$ .

$lim_{x \to - 10} \frac{2 x + 10}{x^{2} + 100 + 2 x^{2} + 20 x}$

Step 1.3.16.3.6

Add $x^{2}$ and $2 x^{2}$ .

$lim_{x \to - 10} \frac{2 x + 10}{3 x^{2} + 100 + 20 x}$

Step 1.3.16.4

Reorder terms.

$lim_{x \to - 10} \frac{2 x + 10}{3 x^{2} + 20 x + 100}$

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 $- 10$ .

$\frac{lim_{x \to - 10} 2 x + 10}{lim_{x \to - 10} 3 x^{2} + 20 x + 100}$

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{lim_{x \to - 10} 2 x + lim_{x \to - 10} 10}{lim_{x \to - 10} 3 x^{2} + 20 x + 100}$

Step 2.3

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

$\frac{2 lim_{x \to - 10} x + lim_{x \to - 10} 10}{lim_{x \to - 10} 3 x^{2} + 20 x + 100}$

Step 2.4

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

$\frac{2 lim_{x \to - 10} x + 10}{lim_{x \to - 10} 3 x^{2} + 20 x + 100}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{2 lim_{x \to - 10} x + 10}{lim_{x \to - 10} 3 x^{2} + lim_{x \to - 10} 20 x + lim_{x \to - 10} 100}$

Step 2.6

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

$\frac{2 lim_{x \to - 10} x + 10}{3 lim_{x \to - 10} x^{2} + lim_{x \to - 10} 20 x + lim_{x \to - 10} 100}$

Step 2.7

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

$\frac{2 lim_{x \to - 10} x + 10}{3 {(lim_{x \to - 10} x)}^{2} + lim_{x \to - 10} 20 x + lim_{x \to - 10} 100}$

Step 2.8

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to - 10} x + 10}{3 {(lim_{x \to - 10} x)}^{2} + 20 lim_{x \to - 10} x + lim_{x \to - 10} 100}$

Step 2.9

Evaluate the limit of $100$ which is constant as $x$ approaches $- 10$ .

$\frac{2 lim_{x \to - 10} x + 10}{3 {(lim_{x \to - 10} x)}^{2} + 20 lim_{x \to - 10} x + 100}$

Step 3

Evaluate the limits by plugging in $- 10$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{2 \cdot - 10 + 10}{3 {(lim_{x \to - 10} x)}^{2} + 20 lim_{x \to - 10} x + 100}$

Step 3.2

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{2 \cdot - 10 + 10}{3 {(- 10)}^{2} + 20 lim_{x \to - 10} x + 100}$

Step 3.3

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{2 \cdot - 10 + 10}{3 {(- 10)}^{2} + 20 \cdot - 10 + 100}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $2$ by $- 10$ .

$\frac{- 20 + 10}{3 {(- 10)}^{2} + 20 \cdot - 10 + 100}$

Step 4.1.2

Add $- 20$ and $10$ .

$\frac{- 10}{3 {(- 10)}^{2} + 20 \cdot - 10 + 100}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Raise $- 10$ to the power of $2$ .

$\frac{- 10}{3 \cdot 100 + 20 \cdot - 10 + 100}$

Step 4.2.2

Multiply $3$ by $100$ .

$\frac{- 10}{300 + 20 \cdot - 10 + 100}$

Step 4.2.3

Multiply $20$ by $- 10$ .

$\frac{- 10}{300 - 200 + 100}$

Step 4.2.4

Subtract $200$ from $300$ .

$\frac{- 10}{100 + 100}$

Step 4.2.5

Add $100$ and $100$ .

$\frac{- 10}{200}$

Step 4.3

Cancel the common factor of $- 10$ and $200$ .

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Step 4.3.1

Factor $10$ out of $- 10$ .

$\frac{10 (- 1)}{200}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $10$ out of $200$ .

$\frac{10 \cdot - 1}{10 \cdot 20}$

Step 4.3.2.2

Cancel the common factor.

$\frac{10 \cdot - 1}{10 \cdot 20}$

Step 4.3.2.3

Rewrite the expression.

$\frac{- 1}{20}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1}{20}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{20}$

Decimal Form:

$- 0.05$