Calculus Examples

$lim_{x \to - \infty} \frac{(x + 2) (x + 5) (x - 5)}{13 (x + 3) (x + 5)}$

Step 1

Cancel the common factor of $x + 5$ .

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Step 1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{(x + 2) (x + 5) (x - 5)}{13 (x + 3) (x + 5)}$

Step 1.2

Rewrite the expression.

$lim_{x \to - \infty} \frac{(x + 2) (x - 5)}{13 (x + 3)}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{x \to - \infty} (x + 2) (x - 5)}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Apply the distributive property.

$\frac{lim_{x \to - \infty} x (x - 5) + 2 (x - 5)}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.2

Apply the distributive property.

$\frac{lim_{x \to - \infty} x \cdot x + x \cdot - 5 + 2 (x - 5)}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.3

Apply the distributive property.

$\frac{lim_{x \to - \infty} x \cdot x + x \cdot - 5 + 2 x + 2 \cdot - 5}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.4

Reorder $x$ and $- 5$ .

$\frac{lim_{x \to - \infty} x \cdot x - 5 \cdot x + 2 x + 2 \cdot - 5}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.5

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

$\frac{lim_{x \to - \infty} x^{1} x - 5 x + 2 x + 2 \cdot - 5}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.6

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

$\frac{lim_{x \to - \infty} x^{1} x^{1} - 5 x + 2 x + 2 \cdot - 5}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.7

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

$\frac{lim_{x \to - \infty} x^{1 + 1} - 5 x + 2 x + 2 \cdot - 5}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.8

Simplify by adding terms.

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Step 2.1.2.8.1

Add $1$ and $1$ .

$\frac{lim_{x \to - \infty} x^{2} - 5 x + 2 x + 2 \cdot - 5}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.8.2

Multiply $2$ by $- 5$ .

$\frac{lim_{x \to - \infty} x^{2} - 5 x + 2 x - 10}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.8.3

Add $- 5 x$ and $2 x$ .

$\frac{lim_{x \to - \infty} x^{2} - 3 x - 10}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.2.9

The limit at negative infinity of a polynomial of even degree whose leading coefficient is positive is infinity.

$\frac{\infty}{lim_{x \to - \infty} 13 (x + 3)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Simplify by multiplying through.

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Step 2.1.3.1.1

Apply the distributive property.

$\frac{\infty}{lim_{x \to - \infty} 13 x + 13 \cdot 3}$

Step 2.1.3.1.2

Multiply $13$ by $3$ .

$\frac{\infty}{lim_{x \to - \infty} 13 x + 39}$

Step 2.1.3.2

The limit at negative infinity of a polynomial of odd degree whose leading coefficient is positive is negative infinity.

$\frac{\infty}{- \infty}$

Step 2.1.3.3

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{- \infty}$

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{- \infty}$

Step 2.2

Since $\frac{\infty}{- \infty}$ 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.

$lim_{x \to - \infty} \frac{(x + 2) (x - 5)}{13 (x + 3)} = lim_{x \to - \infty} \frac{\frac{d}{d x} [(x + 2) (x - 5)]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to - \infty} \frac{\frac{d}{d x} [(x + 2) (x - 5)]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.2

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 $f (x) = x + 2$ and $g (x) = x - 5$ .

$lim_{x \to - \infty} \frac{(x + 2) \frac{d}{d x} [x - 5] + (x - 5) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.3

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

$lim_{x \to - \infty} \frac{(x + 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + (x - 5) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.4

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

$lim_{x \to - \infty} \frac{(x + 2) (1 + \frac{d}{d x} [- 5]) + (x - 5) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$lim_{x \to - \infty} \frac{(x + 2) (1 + 0) + (x - 5) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.6

Add $1$ and $0$ .

$lim_{x \to - \infty} \frac{(x + 2) \cdot 1 + (x - 5) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.7

Multiply $x + 2$ by $1$ .

$lim_{x \to - \infty} \frac{x + 2 + (x - 5) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.8

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

$lim_{x \to - \infty} \frac{x + 2 + (x - 5) (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.9

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

$lim_{x \to - \infty} \frac{x + 2 + (x - 5) (1 + \frac{d}{d x} [2])}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.10

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$lim_{x \to - \infty} \frac{x + 2 + (x - 5) (1 + 0)}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.11

Add $1$ and $0$ .

$lim_{x \to - \infty} \frac{x + 2 + (x - 5) \cdot 1}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.12

Multiply $x - 5$ by $1$ .

$lim_{x \to - \infty} \frac{x + 2 + x - 5}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.13

Add $x$ and $x$ .

$lim_{x \to - \infty} \frac{2 x + 2 - 5}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.14

Subtract $5$ from $2$ .

$lim_{x \to - \infty} \frac{2 x - 3}{\frac{d}{d x} [13 (x + 3)]}$

Step 2.3.15

Since $13$ is constant with respect to $x$ , the derivative of $13 (x + 3)$ with respect to $x$ is $13 \frac{d}{d x} [x + 3]$ .

$lim_{x \to - \infty} \frac{2 x - 3}{13 \frac{d}{d x} [x + 3]}$

Step 2.3.16

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

$lim_{x \to - \infty} \frac{2 x - 3}{13 (\frac{d}{d x} [x] + \frac{d}{d x} [3])}$

Step 2.3.17

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

$lim_{x \to - \infty} \frac{2 x - 3}{13 (1 + \frac{d}{d x} [3])}$

Step 2.3.18

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to - \infty} \frac{2 x - 3}{13 (1 + 0)}$

Step 2.3.19

Add $1$ and $0$ .

$lim_{x \to - \infty} \frac{2 x - 3}{13 \cdot 1}$

Step 2.3.20

Multiply $13$ by $1$ .

$lim_{x \to - \infty} \frac{2 x - 3}{13}$

Step 3

Split the fraction $\frac{2 x - 3}{13}$ into two fractions.

$lim_{x \to - \infty} \frac{2 x}{13} + \frac{- 3}{13}$

Step 4

The limit at negative infinity of a polynomial of odd degree whose leading coefficient is positive is negative infinity.

$- \infty$