Precalculus Examples

$lim_{n \to 8} \frac{(3 n + 4) (1 - n)}{n^{2}}$

Step 1

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

$\frac{lim_{n \to 8} (3 n + 4) (1 - n)}{lim_{n \to 8} n^{2}}$

Step 2

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

$\frac{lim_{n \to 8} 3 n + 4 \cdot lim_{n \to 8} 1 - n}{lim_{n \to 8} n^{2}}$

Step 3

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

$\frac{(lim_{n \to 8} 3 n + lim_{n \to 8} 4) \cdot lim_{n \to 8} 1 - n}{lim_{n \to 8} n^{2}}$

Step 4

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

$\frac{(3 lim_{n \to 8} n + lim_{n \to 8} 4) \cdot lim_{n \to 8} 1 - n}{lim_{n \to 8} n^{2}}$

Step 5

Evaluate the limit of $4$ which is constant as $n$ approaches $8$ .

$\frac{(3 lim_{n \to 8} n + 4) \cdot lim_{n \to 8} 1 - n}{lim_{n \to 8} n^{2}}$

Step 6

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

$\frac{(3 lim_{n \to 8} n + 4) \cdot (lim_{n \to 8} 1 - lim_{n \to 8} n)}{lim_{n \to 8} n^{2}}$

Step 7

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$\frac{(3 lim_{n \to 8} n + 4) \cdot (1 - lim_{n \to 8} n)}{lim_{n \to 8} n^{2}}$

Step 8

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

$\frac{(3 lim_{n \to 8} n + 4) \cdot (1 - lim_{n \to 8} n)}{{(lim_{n \to 8} n)}^{2}}$

Step 9

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

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Step 9.1

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

$\frac{(3 \cdot 8 + 4) \cdot (1 - lim_{n \to 8} n)}{{(lim_{n \to 8} n)}^{2}}$

Step 9.2

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

$\frac{(3 \cdot 8 + 4) \cdot (1 - 8)}{{(lim_{n \to 8} n)}^{2}}$

Step 9.3

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

$\frac{(3 \cdot 8 + 4) \cdot (1 - 8)}{8^{2}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $3$ by $8$ .

$\frac{(24 + 4) (1 - 8)}{8^{2}}$

Step 10.1.2

Add $24$ and $4$ .

$\frac{28 (1 - 8)}{8^{2}}$

Step 10.1.3

Subtract $8$ from $1$ .

$\frac{28 \cdot - 7}{8^{2}}$

Step 10.2

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

$\frac{28 \cdot - 7}{64}$

Step 10.3

Multiply $28$ by $- 7$ .

$\frac{- 196}{64}$

Step 10.4

Cancel the common factor of $- 196$ and $64$ .

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Step 10.4.1

Factor $4$ out of $- 196$ .

$\frac{4 (- 49)}{64}$

Step 10.4.2

Cancel the common factors.

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Step 10.4.2.1

Factor $4$ out of $64$ .

$\frac{4 \cdot - 49}{4 \cdot 16}$

Step 10.4.2.2

Cancel the common factor.

$\frac{4 \cdot - 49}{4 \cdot 16}$

Step 10.4.2.3

Rewrite the expression.

$\frac{- 49}{16}$

Step 10.5

Move the negative in front of the fraction.

$- \frac{49}{16}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{49}{16}$

Decimal Form:

$- 3.0625$