Calculus Examples

$lim_{n \to 8} \frac{n^{3}}{n + 1}$

Step 1

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

$\frac{lim_{n \to 8} n^{3}}{lim_{n \to 8} n + 1}$

Step 2

Move the exponent $3$ from $n^{3}$ outside the limit using the Limits Power Rule.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

$\frac{8^{3}}{lim_{n \to 8} n + 1}$

Step 5.2

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

$\frac{8^{3}}{8 + 1}$

Step 6

Simplify the answer.

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Step 6.1

Raise $8$ to the power of $3$ .

$\frac{512}{8 + 1}$

Step 6.2

Add $8$ and $1$ .

$\frac{512}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{512}{9}$

Decimal Form:

$56.\overline{8}$