Calculus Examples

$lim_{n \to 8} \sqrt[4]{n^{4} - 3 n^{3}} - \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 1

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

$lim_{n \to 8} \sqrt[4]{n^{4} - 3 n^{3}} - lim_{n \to 8} \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 2

Move the limit under the radical sign.

$\sqrt[4]{lim_{n \to 8} n^{4} - 3 n^{3}} - lim_{n \to 8} \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 3

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

$\sqrt[4]{lim_{n \to 8} n^{4} - lim_{n \to 8} 3 n^{3}} - lim_{n \to 8} \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 4

Move the exponent $4$ from $n^{4}$ outside the limit using the Limits Power Rule.

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - lim_{n \to 8} 3 n^{3}} - lim_{n \to 8} \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 5

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

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 lim_{n \to 8} n^{3}} - lim_{n \to 8} \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 6

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

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - lim_{n \to 8} \sqrt[4]{n^{4} + 5 n^{3} + 1}$

Step 7

Move the limit under the radical sign.

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{lim_{n \to 8} n^{4} + 5 n^{3} + 1}$

Step 8

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

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{lim_{n \to 8} n^{4} + lim_{n \to 8} 5 n^{3} + lim_{n \to 8} 1}$

Step 9

Move the exponent $4$ from $n^{4}$ outside the limit using the Limits Power Rule.

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{{(lim_{n \to 8} n)}^{4} + lim_{n \to 8} 5 n^{3} + lim_{n \to 8} 1}$

Step 10

Move the term $5$ outside of the limit because it is constant with respect to $n$ .

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{{(lim_{n \to 8} n)}^{4} + 5 lim_{n \to 8} n^{3} + lim_{n \to 8} 1}$

Step 11

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

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{{(lim_{n \to 8} n)}^{4} + 5 {(lim_{n \to 8} n)}^{3} + lim_{n \to 8} 1}$

Step 12

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

$\sqrt[4]{{(lim_{n \to 8} n)}^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{{(lim_{n \to 8} n)}^{4} + 5 {(lim_{n \to 8} n)}^{3} + 1}$

Step 13

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

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Step 13.1

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

$\sqrt[4]{8^{4} - 3 {(lim_{n \to 8} n)}^{3}} - \sqrt[4]{{(lim_{n \to 8} n)}^{4} + 5 {(lim_{n \to 8} n)}^{3} + 1}$

Step 13.2

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

$\sqrt[4]{8^{4} - 3 \cdot 8^{3}} - \sqrt[4]{{(lim_{n \to 8} n)}^{4} + 5 {(lim_{n \to 8} n)}^{3} + 1}$

Step 13.3

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

$\sqrt[4]{8^{4} - 3 \cdot 8^{3}} - \sqrt[4]{8^{4} + 5 {(lim_{n \to 8} n)}^{3} + 1}$

Step 13.4

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

$\sqrt[4]{8^{4} - 3 \cdot 8^{3}} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14

Simplify each term.

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Step 14.1

Raise $8$ to the power of $4$ .

$\sqrt[4]{4096 - 3 \cdot 8^{3}} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.2

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

$\sqrt[4]{4096 - 3 \cdot 512} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.3

Multiply $- 3$ by $512$ .

$\sqrt[4]{4096 - 1536} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.4

Subtract $1536$ from $4096$ .

$\sqrt[4]{2560} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.5

Rewrite $2560$ as $4^{4} \cdot 10$ .

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Step 14.5.1

Factor $256$ out of $2560$ .

$\sqrt[4]{256 (10)} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.5.2

Rewrite $256$ as $4^{4}$ .

$\sqrt[4]{4^{4} \cdot 10} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.6

Pull terms out from under the radical.

$4 \sqrt[4]{10} - \sqrt[4]{8^{4} + 5 \cdot 8^{3} + 1}$

Step 14.7

Raise $8$ to the power of $4$ .

$4 \sqrt[4]{10} - \sqrt[4]{4096 + 5 \cdot 8^{3} + 1}$

Step 14.8

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

$4 \sqrt[4]{10} - \sqrt[4]{4096 + 5 \cdot 512 + 1}$

Step 14.9

Multiply $5$ by $512$ .

$4 \sqrt[4]{10} - \sqrt[4]{4096 + 2560 + 1}$

Step 14.10

Add $4096$ and $2560$ .

$4 \sqrt[4]{10} - \sqrt[4]{6656 + 1}$

Step 14.11

Add $6656$ and $1$ .

$4 \sqrt[4]{10} - \sqrt[4]{6657}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[4]{10} - \sqrt[4]{6657}$

Decimal Form:

$- 1.91962505 \dots$