Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{x^{3}} + x}{4 x^{2} + 6 x}$

Step 1

Simplify each term.

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Step 1.1

Factor out $x^{2}$ .

$lim_{x \to \infty} \frac{\sqrt{x^{2} x} + x}{4 x^{2} + 6 x}$

Step 1.2

Pull terms out from under the radical.

$lim_{x \to \infty} \frac{x \sqrt{x} + x}{4 x^{2} + 6 x}$

Step 2

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

$lim_{x \to \infty} \frac{\frac{x \sqrt{x}}{x^{2}} + \frac{x}{x^{2}}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Factor $x$ out of $x \sqrt{x}$ .

$lim_{x \to \infty} \frac{\frac{x (\sqrt{x})}{x^{2}} + \frac{x}{x^{2}}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3.1.1.2

Cancel the common factors.

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Step 3.1.1.2.1

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

$lim_{x \to \infty} \frac{\frac{x (\sqrt{x})}{x \cdot x} + \frac{x}{x^{2}}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x \sqrt{x}}{x \cdot x} + \frac{x}{x^{2}}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{\sqrt{x}}{x} + \frac{x}{x^{2}}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3.1.2

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

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Step 3.1.2.1

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

$lim_{x \to \infty} \frac{\frac{\sqrt{x}}{x} + \frac{x^{1}}{x^{2}}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3.1.2.2

Factor $x$ out of $x^{1}$ .

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

Step 3.1.2.3

Cancel the common factors.

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Step 3.1.2.3.1

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

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

Step 3.1.2.3.2

Cancel the common factor.

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

Step 3.1.2.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{\sqrt{x}}{x} + \frac{1}{x}}{\frac{4 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}$

Step 3.2

Simplify each term.

$lim_{x \to \infty} \frac{\frac{\sqrt{x}}{x} + \frac{1}{x}}{4 + \frac{6}{x}}$

Step 3.3

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

$\frac{lim_{x \to \infty} \frac{\sqrt{x}}{x} + \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 3.4

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

$\frac{lim_{x \to \infty} \frac{\sqrt{x}}{x} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 4.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{\frac{lim_{x \to \infty} \sqrt{x}}{lim_{x \to \infty} x} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.1.2

As $x$ approaches $\infty$ for radicals, the value goes to $\infty$ .

$\frac{\frac{\infty}{lim_{x \to \infty} x} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\frac{\infty}{\infty} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\frac{\infty}{\infty} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.2

Since $\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \infty} \frac{\sqrt{x}}{x} = lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{x}]}{\frac{d}{d x} [x]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$\frac{lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{x}]}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.2

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

$\frac{lim_{x \to \infty} \frac{\frac{d}{d x} [x^{\frac{1}{2}}]}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} - 1}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.5

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

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.6

Combine the numerators over the common denominator.

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.7

Simplify the numerator.

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Step 4.3.7.1

Multiply $- 1$ by $2$ .

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1 - 2}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.7.2

Subtract $2$ from $1$ .

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{- 1}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.8

Move the negative in front of the fraction.

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} x^{- \frac{1}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.9

Simplify.

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Step 4.3.9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{lim_{x \to \infty} \frac{\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.9.2

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

$\frac{lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{lim_{x \to \infty} \frac{1}{2 x^{\frac{1}{2}}} \cdot 1 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.5

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

$\frac{lim_{x \to \infty} \frac{1}{2 \sqrt{x}} \cdot 1 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4.6

Multiply $\frac{1}{2 \sqrt{x}}$ by $1$ .

$\frac{lim_{x \to \infty} \frac{1}{2 \sqrt{x}} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 5

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} lim_{x \to \infty} \frac{1}{\sqrt{x}} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 6

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

$\frac{\frac{1}{2} \cdot 0 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 7

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

$\frac{\frac{1}{2} \cdot 0 + 0}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 8

Evaluate the limit.

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Step 8.1

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

$\frac{\frac{1}{2} \cdot 0 + 0}{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{6}{x}}$

Step 8.2

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

$\frac{\frac{1}{2} \cdot 0 + 0}{4 + lim_{x \to \infty} \frac{6}{x}}$

Step 8.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot 0 + 0}{4 + 6 lim_{x \to \infty} \frac{1}{x}}$

Step 9

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

$\frac{\frac{1}{2} \cdot 0 + 0}{4 + 6 \cdot 0}$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $\frac{1}{2} \cdot 0 + 0$ and $4 + 6 \cdot 0$ .

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Step 10.1.1

Factor $2$ out of $\frac{1}{2} \cdot 0$ .

$\frac{2 (\frac{1}{2} \cdot 0) + 0}{4 + 6 \cdot 0}$

Step 10.1.2

Factor $2$ out of $0$ .

$\frac{2 (\frac{1}{2} \cdot 0) + 2 (0)}{4 + 6 \cdot 0}$

Step 10.1.3

Factor $2$ out of $2 (\frac{1}{2} \cdot 0) + 2 (0)$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{4 + 6 \cdot 0}$

Step 10.1.4

Cancel the common factors.

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Step 10.1.4.1

Factor $2$ out of $4$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (2) + 6 \cdot 0}$

Step 10.1.4.2

Factor $2$ out of $6 \cdot 0$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (2) + 2 (3 \cdot 0)}$

Step 10.1.4.3

Factor $2$ out of $2 (2) + 2 (3 \cdot 0)$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (2 + 3 \cdot 0)}$

Step 10.1.4.4

Cancel the common factor.

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (2 + 3 \cdot 0)}$

Step 10.1.4.5

Rewrite the expression.

$\frac{\frac{1}{2} \cdot 0 + 0}{2 + 3 \cdot 0}$

Step 10.2

Cancel the common factor of $\frac{1}{2} \cdot 0 + 0$ and $2 + 3 \cdot 0$ .

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Step 10.2.1

Reorder terms.

$\frac{\frac{1}{2} \cdot 0 + 0}{2 + 0 \cdot 3}$

Step 10.2.2

Factor $2$ out of $\frac{1}{2} \cdot 0$ .

$\frac{2 (\frac{1}{2} \cdot 0) + 0}{2 + 0 \cdot 3}$

Step 10.2.3

Factor $2$ out of $0$ .

$\frac{2 (\frac{1}{2} \cdot 0) + 2 (0)}{2 + 0 \cdot 3}$

Step 10.2.4

Factor $2$ out of $2 (\frac{1}{2} \cdot 0) + 2 (0)$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 + 0 \cdot 3}$

Step 10.2.5

Cancel the common factors.

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Step 10.2.5.1

Factor $2$ out of $2$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (1) + 0 \cdot 3}$

Step 10.2.5.2

Factor $2$ out of $0 \cdot 3$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (1) + 2 (0 \cdot 3)}$

Step 10.2.5.3

Factor $2$ out of $2 (1) + 2 (0 \cdot 3)$ .

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (1 + 0 \cdot 3)}$

Step 10.2.5.4

Cancel the common factor.

$\frac{2 (\frac{1}{2} \cdot 0 + 0)}{2 (1 + 0 \cdot 3)}$

Step 10.2.5.5

Rewrite the expression.

$\frac{\frac{1}{2} \cdot 0 + 0}{1 + 0 \cdot 3}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Multiply $\frac{1}{2}$ by $0$ .

$\frac{0 + 0}{1 + 0 \cdot 3}$

Step 10.3.2

Add $0$ and $0$ .

$\frac{0}{1 + 0 \cdot 3}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Multiply $0$ by $3$ .

$\frac{0}{1 + 0}$

Step 10.4.2

Add $1$ and $0$ .

$\frac{0}{1}$

Step 10.5

Divide $0$ by $1$ .

$0$