Calculus Examples

$lim_{x \to 8} \frac{\sqrt{x^{4} + 8 x^{3} - 9}}{2 x^{2} + 9}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{x^{4} + 8 x^{3} - 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} x^{4} + 8 x^{3} - 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} x^{4} + lim_{x \to 8} 8 x^{3} - lim_{x \to 8} 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 4

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

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 8 x^{3} - lim_{x \to 8} 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 lim_{x \to 8} x^{3} - lim_{x \to 8} 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 6

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

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 7

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 {(lim_{x \to 8} x)}^{3} - 1 \cdot 9}}{lim_{x \to 8} 2 x^{2} + 9}$

Step 8

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

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 {(lim_{x \to 8} x)}^{3} - 1 \cdot 9}}{lim_{x \to 8} 2 x^{2} + lim_{x \to 8} 9}$

Step 9

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

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 {(lim_{x \to 8} x)}^{3} - 1 \cdot 9}}{2 lim_{x \to 8} x^{2} + lim_{x \to 8} 9}$

Step 10

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

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 {(lim_{x \to 8} x)}^{3} - 1 \cdot 9}}{2 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 9}$

Step 11

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{{(lim_{x \to 8} x)}^{4} + 8 {(lim_{x \to 8} x)}^{3} - 1 \cdot 9}}{2 {(lim_{x \to 8} x)}^{2} + 9}$

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 $x$ .

$\frac{\sqrt{8^{4} + 8 {(lim_{x \to 8} x)}^{3} - 1 \cdot 9}}{2 {(lim_{x \to 8} x)}^{2} + 9}$

Step 12.2

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

$\frac{\sqrt{8^{4} + 8 \cdot 8^{3} - 1 \cdot 9}}{2 {(lim_{x \to 8} x)}^{2} + 9}$

Step 12.3

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

$\frac{\sqrt{8^{4} + 8 \cdot 8^{3} - 1 \cdot 9}}{2 \cdot 8^{2} + 9}$

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 $4$ .

$\frac{\sqrt{4096 + 8 \cdot 8^{3} - 1 \cdot 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.2

Multiply $8$ by $8^{3}$ by adding the exponents.

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Step 13.1.2.1

Multiply $8$ by $8^{3}$ .

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Step 13.1.2.1.1

Raise $8$ to the power of $1$ .

$\frac{\sqrt{4096 + 8^{1} \cdot 8^{3} - 1 \cdot 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{4096 + 8^{1 + 3} - 1 \cdot 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.2.2

Add $1$ and $3$ .

$\frac{\sqrt{4096 + 8^{4} - 1 \cdot 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.3

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

$\frac{\sqrt{4096 + 4096 - 1 \cdot 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.4

Multiply $- 1$ by $9$ .

$\frac{\sqrt{4096 + 4096 - 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.5

Add $4096$ and $4096$ .

$\frac{\sqrt{8192 - 9}}{2 \cdot 8^{2} + 9}$

Step 13.1.6

Subtract $9$ from $8192$ .

$\frac{\sqrt{8183}}{2 \cdot 8^{2} + 9}$

Step 13.1.7

Rewrite $8183$ as $7^{2} \cdot 167$ .

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Step 13.1.7.1

Factor $49$ out of $8183$ .

$\frac{\sqrt{49 (167)}}{2 \cdot 8^{2} + 9}$

Step 13.1.7.2

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2} \cdot 167}}{2 \cdot 8^{2} + 9}$

Step 13.1.8

Pull terms out from under the radical.

$\frac{7 \sqrt{167}}{2 \cdot 8^{2} + 9}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{7 \sqrt{167}}{2 \cdot 64 + 9}$

Step 13.2.2

Multiply $2$ by $64$ .

$\frac{7 \sqrt{167}}{128 + 9}$

Step 13.2.3

Add $128$ and $9$ .

$\frac{7 \sqrt{167}}{137}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{167}}{137}$

Decimal Form:

$0.66029150 \dots$