Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $2$ by $64$ .

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

Step 7.1.3

Multiply $- 3$ by $8$ .

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

Step 7.1.4

Multiply $- 1$ by $6$ .

$\frac{128 - 24 - 6}{\sqrt{x^{4} + 1}}$

Step 7.1.5

Subtract $24$ from $128$ .

$\frac{104 - 6}{\sqrt{x^{4} + 1}}$

Step 7.1.6

Subtract $6$ from $104$ .

$\frac{98}{\sqrt{x^{4} + 1}}$

Step 7.2

Multiply $\frac{98}{\sqrt{x^{4} + 1}}$ by $\frac{\sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1}}$ .

$\frac{98}{\sqrt{x^{4} + 1}} \cdot \frac{\sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1}}$

Step 7.3

Combine and simplify the denominator.

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Step 7.3.1

Multiply $\frac{98}{\sqrt{x^{4} + 1}}$ by $\frac{\sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1}}$ .

$\frac{98 \sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1} \sqrt{x^{4} + 1}}$

Step 7.3.2

Raise $\sqrt{x^{4} + 1}$ to the power of $1$ .

$\frac{98 \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{1} \sqrt{x^{4} + 1}}$

Step 7.3.3

Raise $\sqrt{x^{4} + 1}$ to the power of $1$ .

$\frac{98 \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{1} {\sqrt{x^{4} + 1}}^{1}}$

Step 7.3.4

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

$\frac{98 \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{1 + 1}}$

Step 7.3.5

Add $1$ and $1$ .

$\frac{98 \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{2}}$

Step 7.3.6

Rewrite ${\sqrt{x^{4} + 1}}^{2}$ as $x^{4} + 1$ .

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Step 7.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} + 1}$ as ${(x^{4} + 1)}^{\frac{1}{2}}$ .

$\frac{98 \sqrt{x^{4} + 1}}{{({(x^{4} + 1)}^{\frac{1}{2}})}^{2}}$

Step 7.3.6.2

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

$\frac{98 \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 7.3.6.3

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

$\frac{98 \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{\frac{2}{2}}}$

Step 7.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.6.4.1

Cancel the common factor.

$\frac{98 \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{\frac{2}{2}}}$

Step 7.3.6.4.2

Rewrite the expression.

$\frac{98 \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{1}}$

Step 7.3.6.5

Simplify.

$\frac{98 \sqrt{x^{4} + 1}}{x^{4} + 1}$