Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the exponent $3$ from ${(n + 1)}^{3}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Move the exponent $3$ from ${(n + 2)}^{3}$ outside the limit using the Limits Power Rule.

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

Step 9

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

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

Step 10

Evaluate the limit of $2$ which is constant as $n$ approaches $8$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Evaluate the limit of $3$ which is constant as $n$ approaches $8$ .

Step 16

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

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

Step 17

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

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

Step 18

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

Step 19

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

Step 20

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

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Step 20.1