Calculus Examples

$lim_{x \to \infty} \frac{5 + x - 4 x^{2}}{\sqrt{1 + x^{2} + 2 x^{4}}}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2} = \sqrt{x^{4}}$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{x}{x^{2}} + \frac{- 4 x^{2}}{x^{2}}}{\sqrt{\frac{1}{x^{4}} + \frac{x^{2}}{x^{4}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{x^{2}}{x^{4}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 2.2.1.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{x^{2} \cdot 1}{x^{4}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $x^{2}$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{x^{2} \cdot 1}{x^{2} x^{2}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2.2.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{x^{2} \cdot 1}{x^{2} x^{2}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2.2.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2.2.2

Cancel the common factor of $x^{4}$ .

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Step 2.2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + \frac{2 x^{4}}{x^{4}}}}$

Step 2.2.2.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{1}{x} - 4}{\sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \frac{5}{x^{2}} + \frac{1}{x} - 4}{lim_{x \to \infty} \sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{5}{x^{2}} + lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 4}{lim_{x \to \infty} \sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 2.5

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

$\frac{5 lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 4}{lim_{x \to \infty} \sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{5 \cdot 0 + lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 4}{lim_{x \to \infty} \sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{5 \cdot 0 + 0 - lim_{x \to \infty} 4}{lim_{x \to \infty} \sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{5 \cdot 0 + 0 - 1 \cdot 4}{lim_{x \to \infty} \sqrt{\frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 5.2

Move the limit under the radical sign.

$\frac{5 \cdot 0 + 0 - 1 \cdot 4}{\sqrt{lim_{x \to \infty} \frac{1}{x^{4}} + \frac{1}{x^{2}} + 2}}$

Step 5.3

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

$\frac{5 \cdot 0 + 0 - 1 \cdot 4}{\sqrt{lim_{x \to \infty} \frac{1}{x^{4}} + lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 2}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

$\frac{5 \cdot 0 + 0 - 1 \cdot 4}{\sqrt{0 + lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 2}}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{5 \cdot 0 + 0 - 1 \cdot 4}{\sqrt{0 + 0 + lim_{x \to \infty} 2}}$

Step 8

Evaluate the limit.

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Step 8.1

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

$\frac{5 \cdot 0 + 0 - 1 \cdot 4}{\sqrt{0 + 0 + 2}}$

Step 8.2

Simplify the answer.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Multiply $5$ by $0$ .

$\frac{0 + 0 - 1 \cdot 4}{\sqrt{0 + 0 + 2}}$

Step 8.2.1.2

Multiply $- 1$ by $4$ .

$\frac{0 + 0 - 4}{\sqrt{0 + 0 + 2}}$

Step 8.2.1.3

Add $0$ and $0$ .

$\frac{0 - 4}{\sqrt{0 + 0 + 2}}$

Step 8.2.1.4

Subtract $4$ from $0$ .

$\frac{- 4}{\sqrt{0 + 0 + 2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Add $0$ and $0$ .

$\frac{- 4}{\sqrt{0 + 2}}$

Step 8.2.2.2

Add $0$ and $2$ .

$\frac{- 4}{\sqrt{2}}$

Step 8.2.3

Move the negative in front of the fraction.

$- \frac{4}{\sqrt{2}}$

Step 8.2.4

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- (\frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 8.2.5

Combine and simplify the denominator.

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Step 8.2.5.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 8.2.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{4 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 8.2.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{4 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 8.2.5.4

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

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

Step 8.2.5.5

Add $1$ and $1$ .

$- \frac{4 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 8.2.5.6

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

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Step 8.2.5.6.1

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

$- \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 8.2.5.6.2

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

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

Step 8.2.5.6.3

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

$- \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 8.2.5.6.4

Cancel the common factor of $2$ .

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Step 8.2.5.6.4.1

Cancel the common factor.

$- \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 8.2.5.6.4.2

Rewrite the expression.

$- \frac{4 \sqrt{2}}{2^{1}}$

Step 8.2.5.6.5

Evaluate the exponent.

$- \frac{4 \sqrt{2}}{2}$

Step 8.2.6

Cancel the common factor of $4$ and $2$ .

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Step 8.2.6.1

Factor $2$ out of $4 \sqrt{2}$ .

$- \frac{2 (2 \sqrt{2})}{2}$

Step 8.2.6.2

Cancel the common factors.

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Step 8.2.6.2.1

Factor $2$ out of $2$ .

$- \frac{2 (2 \sqrt{2})}{2 (1)}$

Step 8.2.6.2.2

Cancel the common factor.

$- \frac{2 (2 \sqrt{2})}{2 \cdot 1}$

Step 8.2.6.2.3

Rewrite the expression.

$- \frac{2 \sqrt{2}}{1}$

Step 8.2.6.2.4

Divide $2 \sqrt{2}$ by $1$ .

$- (2 \sqrt{2})$

Step 8.2.7

Multiply $2$ by $- 1$ .

$- 2 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{2}$

Decimal Form:

$- 2.82842712 \dots$