Calculus Examples

$lim_{x \to 8} \frac{\sqrt{x^{4} - 3 x^{3} - 1}}{1 x^{2} + 2}$

Step 1

Multiply $x^{2}$ by $1$ .

$lim_{x \to 8} \frac{\sqrt{x^{4} - 3 x^{3} - 1}}{x^{2} + 2}$

Step 2

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

$\frac{lim_{x \to 8} \sqrt{x^{4} - 3 x^{3} - 1}}{lim_{x \to 8} x^{2} + 2}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} x^{4} - 3 x^{3} - 1}}{lim_{x \to 8} x^{2} + 2}$

Step 4

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

$\frac{\sqrt{lim_{x \to 8} x^{4} - lim_{x \to 8} 3 x^{3} - lim_{x \to 8} 1}}{lim_{x \to 8} x^{2} + 2}$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{\sqrt{4096 - 3 \cdot 8^{3} - 1 \cdot 1}}{8^{2} + 2}$

Step 13.1.2

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

$\frac{\sqrt{4096 - 3 \cdot 512 - 1 \cdot 1}}{8^{2} + 2}$

Step 13.1.3

Multiply $- 3$ by $512$ .

$\frac{\sqrt{4096 - 1536 - 1 \cdot 1}}{8^{2} + 2}$

Step 13.1.4

Multiply $- 1$ by $1$ .

$\frac{\sqrt{4096 - 1536 - 1}}{8^{2} + 2}$

Step 13.1.5

Subtract $1536$ from $4096$ .

$\frac{\sqrt{2560 - 1}}{8^{2} + 2}$

Step 13.1.6

Subtract $1$ from $2560$ .

$\frac{\sqrt{2559}}{8^{2} + 2}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{\sqrt{2559}}{64 + 2}$

Step 13.2.2

Add $64$ and $2$ .

$\frac{\sqrt{2559}}{66}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2559}}{66}$

Decimal Form:

$0.76646302 \dots$