Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

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

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Simplify terms.

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Step 2.2.1

Simplify each term.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Cancel the common factors.

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Step 2.2.2.3.1

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

$lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 1}{x \cdot x^{3}}} + 1}{\frac{5}{x} - 1}$

Step 2.2.2.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 1}{x \cdot x^{3}}} + 1}{\frac{5}{x} - 1}$

Step 2.2.2.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{\frac{1}{x^{3}}} + 1}{\frac{5}{x} - 1}$

Step 2.3

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

$\frac{lim_{x \to \infty} \sqrt{\frac{1}{x^{3}}} + 1}{lim_{x \to \infty} \frac{5}{x} - 1}$

Step 2.4

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

$\frac{lim_{x \to \infty} \sqrt{\frac{1}{x^{3}}} + lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{5}{x} - 1}$

Step 2.5

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} \frac{1}{x^{3}}} + lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{5}{x} - 1}$

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Rewrite $0$ as $0^{2}$ .

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

Step 6.2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0 + 1}{5 \cdot 0 - 1 \cdot 1}$

Step 6.2.1.3

Add $0$ and $1$ .

$\frac{1}{5 \cdot 0 - 1 \cdot 1}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $5$ by $0$ .

$\frac{1}{0 - 1 \cdot 1}$

Step 6.2.2.2

Multiply $- 1$ by $1$ .

$\frac{1}{0 - 1}$

Step 6.2.2.3

Subtract $1$ from $0$ .

$\frac{1}{- 1}$

Step 6.2.3

Divide $1$ by $- 1$ .

$- 1$