Calculus Examples

$lim_{n \to - 8} \frac{2 a n}{3 n^{2} - 3 n - 7}$

Step 1

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

$2 a lim_{n \to - 8} \frac{n}{3 n^{2} - 3 n - 7}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $- 8$ .

$2 a \frac{lim_{n \to - 8} n}{lim_{n \to - 8} 3 n^{2} - 3 n - 7}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $- 8$ .

$2 a \frac{lim_{n \to - 8} n}{lim_{n \to - 8} 3 n^{2} - lim_{n \to - 8} 3 n - lim_{n \to - 8} 7}$

Step 4

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

$2 a \frac{lim_{n \to - 8} n}{3 lim_{n \to - 8} n^{2} - lim_{n \to - 8} 3 n - lim_{n \to - 8} 7}$

Step 5

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

$2 a \frac{lim_{n \to - 8} n}{3 {(lim_{n \to - 8} n)}^{2} - lim_{n \to - 8} 3 n - lim_{n \to - 8} 7}$

Step 6

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

$2 a \frac{lim_{n \to - 8} n}{3 {(lim_{n \to - 8} n)}^{2} - 3 lim_{n \to - 8} n - lim_{n \to - 8} 7}$

Step 7

Evaluate the limit of $7$ which is constant as $n$ approaches $- 8$ .

$2 a \frac{lim_{n \to - 8} n}{3 {(lim_{n \to - 8} n)}^{2} - 3 lim_{n \to - 8} n - 1 \cdot 7}$

Step 8

Evaluate the limits by plugging in $- 8$ for all occurrences of $n$ .

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Step 8.1

Evaluate the limit of $n$ by plugging in $- 8$ for $n$ .

$2 a \frac{- 8}{3 {(lim_{n \to - 8} n)}^{2} - 3 lim_{n \to - 8} n - 1 \cdot 7}$

Step 8.2

Evaluate the limit of $n$ by plugging in $- 8$ for $n$ .

$2 a \frac{- 8}{3 {(- 8)}^{2} - 3 lim_{n \to - 8} n - 1 \cdot 7}$

Step 8.3

Evaluate the limit of $n$ by plugging in $- 8$ for $n$ .

$2 a \frac{- 8}{3 {(- 8)}^{2} - 3 \cdot - 8 - 1 \cdot 7}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Raise $- 8$ to the power of $2$ .

$2 a \frac{- 8}{3 \cdot 64 - 3 \cdot - 8 - 1 \cdot 7}$

Step 9.1.2

Multiply $3$ by $64$ .

$2 a \frac{- 8}{192 - 3 \cdot - 8 - 1 \cdot 7}$

Step 9.1.3

Multiply $- 3$ by $- 8$ .

$2 a \frac{- 8}{192 + 24 - 1 \cdot 7}$

Step 9.1.4

Multiply $- 1$ by $7$ .

$2 a \frac{- 8}{192 + 24 - 7}$

Step 9.1.5

Add $192$ and $24$ .

$2 a \frac{- 8}{216 - 7}$

Step 9.1.6

Subtract $7$ from $216$ .

$2 a \frac{- 8}{209}$

Step 9.2

Move the negative in front of the fraction.

$2 a (- \frac{8}{209})$

Step 9.3

Multiply $2 a (- \frac{8}{209})$ .

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Step 9.3.1

Multiply $- 1$ by $2$ .

$- 2 a \frac{8}{209}$

Step 9.3.2

Combine $- 2$ and $\frac{8}{209}$ .

$a \frac{- 2 \cdot 8}{209}$

Step 9.3.3

Multiply $- 2$ by $8$ .

$a \frac{- 16}{209}$

Step 9.3.4

Combine $a$ and $\frac{- 16}{209}$ .

$\frac{a \cdot - 16}{209}$

Step 9.4

Move $- 16$ to the left of $a$ .

$\frac{- 16 \cdot a}{209}$

Step 9.5

Move the negative in front of the fraction.

$- \frac{16 a}{209}$