Calculus Examples

$lim_{n \to - 5} \frac{\frac{2}{n + 3} - \frac{3}{n + 2}}{n + 5}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{2}{n + 3}$ as a fraction with a common denominator, multiply by $\frac{n + 2}{n + 2}$ .

$lim_{n \to - 5} \frac{\frac{2}{n + 3} \cdot \frac{n + 2}{n + 2} - \frac{3}{n + 2}}{n + 5}$

Step 1.2

To write $- \frac{3}{n + 2}$ as a fraction with a common denominator, multiply by $\frac{n + 3}{n + 3}$ .

$lim_{n \to - 5} \frac{\frac{2}{n + 3} \cdot \frac{n + 2}{n + 2} - \frac{3}{n + 2} \cdot \frac{n + 3}{n + 3}}{n + 5}$

Step 1.3

Write each expression with a common denominator of $(n + 3) (n + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2}{n + 3}$ by $\frac{n + 2}{n + 2}$ .

$lim_{n \to - 5} \frac{\frac{2 (n + 2)}{(n + 3) (n + 2)} - \frac{3}{n + 2} \cdot \frac{n + 3}{n + 3}}{n + 5}$

Step 1.3.2

Multiply $\frac{3}{n + 2}$ by $\frac{n + 3}{n + 3}$ .

$lim_{n \to - 5} \frac{\frac{2 (n + 2)}{(n + 3) (n + 2)} - \frac{3 (n + 3)}{(n + 2) (n + 3)}}{n + 5}$

Step 1.3.3

Reorder the factors of $(n + 3) (n + 2)$ .

$lim_{n \to - 5} \frac{\frac{2 (n + 2)}{(n + 2) (n + 3)} - \frac{3 (n + 3)}{(n + 2) (n + 3)}}{n + 5}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{n \to - 5} \frac{\frac{2 (n + 2) - 3 (n + 3)}{(n + 2) (n + 3)}}{n + 5}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{n \to - 5} \frac{2 (n + 2) - 3 (n + 3)}{(n + 2) (n + 3)} \cdot \frac{1}{n + 5}$

Step 2.2

Multiply $\frac{2 (n + 2) - 3 (n + 3)}{(n + 2) (n + 3)}$ by $\frac{1}{n + 5}$ .

$lim_{n \to - 5} \frac{2 (n + 2) - 3 (n + 3)}{(n + 2) (n + 3) (n + 5)}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$\frac{lim_{n \to - 5} 2 (n + 2) - 3 (n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

$\frac{lim_{n \to - 5} 2 (n + 2) - lim_{n \to - 5} 3 (n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.2

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

$\frac{2 lim_{n \to - 5} n + 2 - lim_{n \to - 5} 3 (n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.3

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

$\frac{2 (lim_{n \to - 5} n + lim_{n \to - 5} 2) - lim_{n \to - 5} 3 (n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.4

Evaluate the limit of $2$ which is constant as $n$ approaches $- 5$ .

$\frac{2 (lim_{n \to - 5} n + 2) - lim_{n \to - 5} 3 (n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.5

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

$\frac{2 (lim_{n \to - 5} n + 2) - 3 lim_{n \to - 5} n + 3}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.6

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

$\frac{2 (lim_{n \to - 5} n + 2) - 3 (lim_{n \to - 5} n + lim_{n \to - 5} 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.7

Evaluate the limit of $3$ which is constant as $n$ approaches $- 5$ .

$\frac{2 (lim_{n \to - 5} n + 2) - 3 (lim_{n \to - 5} n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.8

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

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Step 3.1.2.8.1

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

$\frac{2 (- 5 + 2) - 3 (lim_{n \to - 5} n + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.8.2

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

$\frac{2 (- 5 + 2) - 3 (- 5 + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.9

Simplify the answer.

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Step 3.1.2.9.1

Simplify each term.

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Step 3.1.2.9.1.1

Add $- 5$ and $2$ .

$\frac{2 \cdot - 3 - 3 (- 5 + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.9.1.2

Multiply $2$ by $- 3$ .

$\frac{- 6 - 3 (- 5 + 3)}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.9.1.3

Add $- 5$ and $3$ .

$\frac{- 6 - 3 \cdot - 2}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.9.1.4

Multiply $- 3$ by $- 2$ .

$\frac{- 6 + 6}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.2.9.2

Add $- 6$ and $6$ .

$\frac{0}{lim_{n \to - 5} (n + 2) (n + 3) (n + 5)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $- 5$ .

$\frac{0}{lim_{n \to - 5} n + 2 \cdot lim_{n \to - 5} n + 3 \cdot lim_{n \to - 5} n + 5}$

Step 3.1.3.2

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

$\frac{0}{(lim_{n \to - 5} n + lim_{n \to - 5} 2) \cdot lim_{n \to - 5} n + 3 \cdot lim_{n \to - 5} n + 5}$

Step 3.1.3.3

Evaluate the limit of $2$ which is constant as $n$ approaches $- 5$ .

$\frac{0}{(lim_{n \to - 5} n + 2) \cdot lim_{n \to - 5} n + 3 \cdot lim_{n \to - 5} n + 5}$

Step 3.1.3.4

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

$\frac{0}{(lim_{n \to - 5} n + 2) \cdot (lim_{n \to - 5} n + lim_{n \to - 5} 3) \cdot lim_{n \to - 5} n + 5}$

Step 3.1.3.5

Evaluate the limit of $3$ which is constant as $n$ approaches $- 5$ .

$\frac{0}{(lim_{n \to - 5} n + 2) \cdot (lim_{n \to - 5} n + 3) \cdot lim_{n \to - 5} n + 5}$

Step 3.1.3.6

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

$\frac{0}{(lim_{n \to - 5} n + 2) \cdot (lim_{n \to - 5} n + 3) \cdot (lim_{n \to - 5} n + lim_{n \to - 5} 5)}$

Step 3.1.3.7

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

$\frac{0}{(lim_{n \to - 5} n + 2) \cdot (lim_{n \to - 5} n + 3) \cdot (lim_{n \to - 5} n + 5)}$

Step 3.1.3.8

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

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Step 3.1.3.8.1

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

$\frac{0}{(- 5 + 2) \cdot (lim_{n \to - 5} n + 3) \cdot (lim_{n \to - 5} n + 5)}$

Step 3.1.3.8.2

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

$\frac{0}{(- 5 + 2) \cdot (- 5 + 3) \cdot (lim_{n \to - 5} n + 5)}$

Step 3.1.3.8.3

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

$\frac{0}{(- 5 + 2) \cdot (- 5 + 3) \cdot (- 5 + 5)}$

Step 3.1.3.9

Simplify the answer.

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Step 3.1.3.9.1

Add $- 5$ and $2$ .

$\frac{0}{- 3 \cdot (- 5 + 3) \cdot (- 5 + 5)}$

Step 3.1.3.9.2

Add $- 5$ and $3$ .

$\frac{0}{- 3 \cdot - 2 \cdot (- 5 + 5)}$

Step 3.1.3.9.3

Multiply $- 3$ by $- 2$ .

$\frac{0}{6 \cdot (- 5 + 5)}$

Step 3.1.3.9.4

Add $- 5$ and $5$ .

$\frac{0}{6 \cdot 0}$

Step 3.1.3.9.5

Multiply $6$ by $0$ .

$\frac{0}{0}$

Step 3.1.3.9.6

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

Undefined

$\frac{0}{0}$

Step 3.1.3.10

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

By the Sum Rule, the derivative of $2 (n + 2) - 3 (n + 3)$ with respect to $n$ is $\frac{d}{d n} [2 (n + 2)] + \frac{d}{d n} [- 3 (n + 3)]$ .

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

Step 3.3.3

Evaluate $\frac{d}{d n} [2 (n + 2)]$ .

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Step 3.3.3.1

Since $2$ is constant with respect to $n$ , the derivative of $2 (n + 2)$ with respect to $n$ is $2 \frac{d}{d n} [n + 2]$ .

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.3.4

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

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

Step 3.3.3.5

Add $1$ and $0$ .

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

Step 3.3.3.6

Multiply $2$ by $1$ .

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

Step 3.3.4

Evaluate $\frac{d}{d n} [- 3 (n + 3)]$ .

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Step 3.3.4.1

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

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 3.3.4.4

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

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

Step 3.3.4.5

Add $1$ and $0$ .

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

Step 3.3.4.6

Multiply $- 3$ by $1$ .

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

Step 3.3.5

Subtract $3$ from $2$ .

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

Step 3.3.6

Differentiate using the Product Rule which states that $\frac{d}{d n} [f (n) g (n)]$ is $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ where $f (n) = (n + 2) (n + 3)$ and $g (n) = n + 5$ .

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

Add $1$ and $0$ .

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

Step 3.3.11

Multiply $n + 2$ by $1$ .

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

Step 3.3.12

Differentiate using the Product Rule which states that $\frac{d}{d n} [f (n) g (n)]$ is $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ where $f (n) = n + 2$ and $g (n) = n + 3$ .

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

Step 3.3.13

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

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

Step 3.3.14

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

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

Step 3.3.15

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

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

Step 3.3.16

Add $1$ and $0$ .

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

Step 3.3.17

Multiply $n + 2$ by $1$ .

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

Step 3.3.18

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

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

Step 3.3.19

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

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

Step 3.3.20

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

$lim_{n \to - 5} \frac{- 1}{(n + 2) (n + 3) + (n + 5) (n + 2 + (n + 3) (1 + 0))}$

Step 3.3.21

Add $1$ and $0$ .

$lim_{n \to - 5} \frac{- 1}{(n + 2) (n + 3) + (n + 5) (n + 2 + (n + 3) \cdot 1)}$

Step 3.3.22

Multiply $n + 3$ by $1$ .

$lim_{n \to - 5} \frac{- 1}{(n + 2) (n + 3) + (n + 5) (n + 2 + n + 3)}$

Step 3.3.23

Add $n$ and $n$ .

$lim_{n \to - 5} \frac{- 1}{(n + 2) (n + 3) + (n + 5) (2 n + 2 + 3)}$

Step 3.3.24

Add $2$ and $3$ .

$lim_{n \to - 5} \frac{- 1}{(n + 2) (n + 3) + (n + 5) (2 n + 5)}$

Step 3.3.25

Simplify.

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Step 3.3.25.1

Apply the distributive property.

$lim_{n \to - 5} \frac{- 1}{(n + 2) n + (n + 2) \cdot 3 + (n + 5) (2 n + 5)}$

Step 3.3.25.2

Apply the distributive property.

$lim_{n \to - 5} \frac{- 1}{n \cdot n + 2 n + (n + 2) \cdot 3 + (n + 5) (2 n + 5)}$

Step 3.3.25.3

Apply the distributive property.

$lim_{n \to - 5} \frac{- 1}{n \cdot n + 2 n + n \cdot 3 + 2 \cdot 3 + (n + 5) (2 n + 5)}$

Step 3.3.25.4

Apply the distributive property.

$lim_{n \to - 5} \frac{- 1}{n \cdot n + 2 n + n \cdot 3 + 2 \cdot 3 + (n + 5) (2 n) + (n + 5) \cdot 5}$

Step 3.3.25.5

Apply the distributive property.

$lim_{n \to - 5} \frac{- 1}{n \cdot n + 2 n + n \cdot 3 + 2 \cdot 3 + n (2 n) + 5 (2 n) + (n + 5) \cdot 5}$

Step 3.3.25.6

Apply the distributive property.

$lim_{n \to - 5} \frac{- 1}{n \cdot n + 2 n + n \cdot 3 + 2 \cdot 3 + n (2 n) + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7

Combine terms.

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Step 3.3.25.7.1

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

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

Step 3.3.25.7.2

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

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

Step 3.3.25.7.3

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

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

Step 3.3.25.7.4

Add $1$ and $1$ .

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

Step 3.3.25.7.5

Move $3$ to the left of $n$ .

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

Step 3.3.25.7.6

Multiply $2$ by $3$ .

$lim_{n \to - 5} \frac{- 1}{n^{2} + 2 n + 3 n + 6 + n (2 n) + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.7

Add $2 n$ and $3 n$ .

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + n (2 n) + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.8

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

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 (n^{1} n) + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.9

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

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 (n^{1} n^{1}) + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.10

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

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 n^{1 + 1} + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.11

Add $1$ and $1$ .

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 n^{2} + 5 (2 n) + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.12

Multiply $2$ by $5$ .

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 n^{2} + 10 n + n \cdot 5 + 5 \cdot 5}$

Step 3.3.25.7.13

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

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 n^{2} + 10 n + 5 \cdot n + 5 \cdot 5}$

Step 3.3.25.7.14

Multiply $5$ by $5$ .

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 n^{2} + 10 n + 5 n + 25}$

Step 3.3.25.7.15

Add $10 n$ and $5 n$ .

$lim_{n \to - 5} \frac{- 1}{n^{2} + 5 n + 6 + 2 n^{2} + 15 n + 25}$

Step 3.3.25.7.16

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

$lim_{n \to - 5} \frac{- 1}{3 n^{2} + 5 n + 6 + 15 n + 25}$

Step 3.3.25.7.17

Add $5 n$ and $15 n$ .

$lim_{n \to - 5} \frac{- 1}{3 n^{2} + 20 n + 6 + 25}$

Step 3.3.25.7.18

Add $6$ and $25$ .

$lim_{n \to - 5} \frac{- 1}{3 n^{2} + 20 n + 31}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{lim_{n \to - 5} - 1}{lim_{n \to - 5} 3 n^{2} + 20 n + 31}$

Step 4.2

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

$\frac{- 1}{lim_{n \to - 5} 3 n^{2} + 20 n + 31}$

Step 4.3

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

$\frac{- 1}{lim_{n \to - 5} 3 n^{2} + lim_{n \to - 5} 20 n + lim_{n \to - 5} 31}$

Step 4.4

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

$\frac{- 1}{3 lim_{n \to - 5} n^{2} + lim_{n \to - 5} 20 n + lim_{n \to - 5} 31}$

Step 4.5

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

$\frac{- 1}{3 {(lim_{n \to - 5} n)}^{2} + lim_{n \to - 5} 20 n + lim_{n \to - 5} 31}$

Step 4.6

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

$\frac{- 1}{3 {(lim_{n \to - 5} n)}^{2} + 20 lim_{n \to - 5} n + lim_{n \to - 5} 31}$

Step 4.7

Evaluate the limit of $31$ which is constant as $n$ approaches $- 5$ .

$\frac{- 1}{3 {(lim_{n \to - 5} n)}^{2} + 20 lim_{n \to - 5} n + 31}$

Step 5

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

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Step 5.1

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

$\frac{- 1}{3 {(- 5)}^{2} + 20 lim_{n \to - 5} n + 31}$

Step 5.2

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

$\frac{- 1}{3 {(- 5)}^{2} + 20 \cdot - 5 + 31}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

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

$\frac{- 1}{3 \cdot 25 + 20 \cdot - 5 + 31}$

Step 6.1.2

Multiply $3$ by $25$ .

$\frac{- 1}{75 + 20 \cdot - 5 + 31}$

Step 6.1.3

Multiply $20$ by $- 5$ .

$\frac{- 1}{75 - 100 + 31}$

Step 6.1.4

Subtract $100$ from $75$ .

$\frac{- 1}{- 25 + 31}$

Step 6.1.5

Add $- 25$ and $31$ .

$\frac{- 1}{6}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{1}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{6}$

Decimal Form:

$- 0.1\overline{6}$