Precalculus Examples

$lim_{n \to 8} \frac{n^{2} + 1}{n^{2}}$

Step 1

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

$\frac{lim_{n \to 8} n^{2} + 1}{lim_{n \to 8} n^{2}}$

Step 2

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

$\frac{lim_{n \to 8} n^{2} + lim_{n \to 8} 1}{lim_{n \to 8} n^{2}}$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

$\frac{8^{2} + 1}{8^{2}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{64 + 1}{8^{2}}$

Step 7.1.2

Add $64$ and $1$ .

$\frac{65}{8^{2}}$

Step 7.2

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

$\frac{65}{64}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{65}{64}$

Decimal Form:

$1.015625$