Calculus Examples

$lim_{x \to 8} \frac{2 x^{2} - 3 x - 6}{\sqrt{x^{4} - 1}}$

Step 1

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

$\frac{lim_{x \to 8} 2 x^{2} - 3 x - 6}{lim_{x \to 8} \sqrt{x^{4} - 1}}$

Step 2

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

$\frac{lim_{x \to 8} 2 x^{2} - lim_{x \to 8} 3 x - lim_{x \to 8} 6}{lim_{x \to 8} \sqrt{x^{4} - 1}}$

Step 3

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

$\frac{2 lim_{x \to 8} x^{2} - lim_{x \to 8} 3 x - lim_{x \to 8} 6}{lim_{x \to 8} \sqrt{x^{4} - 1}}$

Step 4

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

$\frac{2 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 3 x - lim_{x \to 8} 6}{lim_{x \to 8} \sqrt{x^{4} - 1}}$

Step 5

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

$\frac{2 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - lim_{x \to 8} 6}{lim_{x \to 8} \sqrt{x^{4} - 1}}$

Step 6

Evaluate the limit of $6$ which is constant as $x$ approaches $8$ .

$\frac{2 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}{lim_{x \to 8} \sqrt{x^{4} - 1}}$

Step 7

Move the limit under the radical sign.

$\frac{2 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}{\sqrt{lim_{x \to 8} x^{4} - 1}}$

Step 8

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

$\frac{2 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}{\sqrt{lim_{x \to 8} x^{4} - lim_{x \to 8} 1}}$

Step 9

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

$\frac{2 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}{\sqrt{{(lim_{x \to 8} x)}^{4} - lim_{x \to 8} 1}}$

Step 10

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

$\frac{2 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}{\sqrt{{(lim_{x \to 8} x)}^{4} - 1 \cdot 1}}$

Step 11

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

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Step 11.1

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

$\frac{2 \cdot 8^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}{\sqrt{{(lim_{x \to 8} x)}^{4} - 1 \cdot 1}}$

Step 11.2

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

$\frac{2 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}{\sqrt{{(lim_{x \to 8} x)}^{4} - 1 \cdot 1}}$

Step 11.3

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

$\frac{2 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

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

$\frac{2 \cdot 64 - 3 \cdot 8 - 1 \cdot 6}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12.1.2

Multiply $2$ by $64$ .

$\frac{128 - 3 \cdot 8 - 1 \cdot 6}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12.1.3

Multiply $- 3$ by $8$ .

$\frac{128 - 24 - 1 \cdot 6}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12.1.4

Multiply $- 1$ by $6$ .

$\frac{128 - 24 - 6}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12.1.5

Subtract $24$ from $128$ .

$\frac{104 - 6}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12.1.6

Subtract $6$ from $104$ .

$\frac{98}{\sqrt{8^{4} - 1 \cdot 1}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

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

$\frac{98}{\sqrt{4096 - 1 \cdot 1}}$

Step 12.2.2

Multiply $- 1$ by $1$ .

$\frac{98}{\sqrt{4096 - 1}}$

Step 12.2.3

Subtract $1$ from $4096$ .

$\frac{98}{\sqrt{4095}}$

Step 12.2.4

Rewrite $4095$ as $3^{2} \cdot 455$ .

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Step 12.2.4.1

Factor $9$ out of $4095$ .

$\frac{98}{\sqrt{9 (455)}}$

Step 12.2.4.2

Rewrite $9$ as $3^{2}$ .

$\frac{98}{\sqrt{3^{2} \cdot 455}}$

Step 12.2.5

Pull terms out from under the radical.

$\frac{98}{3 \sqrt{455}}$

Step 12.3

Multiply $\frac{98}{3 \sqrt{455}}$ by $\frac{\sqrt{455}}{\sqrt{455}}$ .

$\frac{98}{3 \sqrt{455}} \cdot \frac{\sqrt{455}}{\sqrt{455}}$

Step 12.4

Combine and simplify the denominator.

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Step 12.4.1

Multiply $\frac{98}{3 \sqrt{455}}$ by $\frac{\sqrt{455}}{\sqrt{455}}$ .

$\frac{98 \sqrt{455}}{3 \sqrt{455} \sqrt{455}}$

Step 12.4.2

Move $\sqrt{455}$ .

$\frac{98 \sqrt{455}}{3 (\sqrt{455} \sqrt{455})}$

Step 12.4.3

Raise $\sqrt{455}$ to the power of $1$ .

$\frac{98 \sqrt{455}}{3 ({\sqrt{455}}^{1} \sqrt{455})}$

Step 12.4.4

Raise $\sqrt{455}$ to the power of $1$ .

$\frac{98 \sqrt{455}}{3 ({\sqrt{455}}^{1} {\sqrt{455}}^{1})}$

Step 12.4.5

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

$\frac{98 \sqrt{455}}{3 {\sqrt{455}}^{1 + 1}}$

Step 12.4.6

Add $1$ and $1$ .

$\frac{98 \sqrt{455}}{3 {\sqrt{455}}^{2}}$

Step 12.4.7

Rewrite ${\sqrt{455}}^{2}$ as $455$ .

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Step 12.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{455}$ as $455^{\frac{1}{2}}$ .

$\frac{98 \sqrt{455}}{3 {(455^{\frac{1}{2}})}^{2}}$

Step 12.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{98 \sqrt{455}}{3 \cdot 455^{\frac{1}{2} \cdot 2}}$

Step 12.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{98 \sqrt{455}}{3 \cdot 455^{\frac{2}{2}}}$

Step 12.4.7.4

Cancel the common factor of $2$ .

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Step 12.4.7.4.1

Cancel the common factor.

$\frac{98 \sqrt{455}}{3 \cdot 455^{\frac{2}{2}}}$

Step 12.4.7.4.2

Rewrite the expression.

$\frac{98 \sqrt{455}}{3 \cdot 455^{1}}$

Step 12.4.7.5

Evaluate the exponent.

$\frac{98 \sqrt{455}}{3 \cdot 455}$

Step 12.5

Cancel the common factor of $98$ and $455$ .

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Step 12.5.1

Factor $7$ out of $98 \sqrt{455}$ .

$\frac{7 (14 \sqrt{455})}{3 \cdot 455}$

Step 12.5.2

Cancel the common factors.

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Step 12.5.2.1

Factor $7$ out of $3 \cdot 455$ .

$\frac{7 (14 \sqrt{455})}{7 (3 \cdot 65)}$

Step 12.5.2.2

Cancel the common factor.

$\frac{7 (14 \sqrt{455})}{7 (3 \cdot 65)}$

Step 12.5.2.3

Rewrite the expression.

$\frac{14 \sqrt{455}}{3 \cdot 65}$

Step 12.6

Multiply $3$ by $65$ .

$\frac{14 \sqrt{455}}{195}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{14 \sqrt{455}}{195}$

Decimal Form:

$1.53143695 \dots$