Calculus Examples

$f (x) = \frac{\sqrt{x}}{x - 4 \sqrt{x} + 4}$

Step 1

Find where the expression $\frac{\sqrt{x}}{x - 4 \sqrt{x} + 4}$ is undefined.

$x < 0, x = 4$

Step 2

Since $\frac{\sqrt{x}}{x - 4 \sqrt{x} + 4}$ $\to$ $\infty$ as $x$ $\to$ $4$ from the left and $\frac{\sqrt{x}}{x - 4 \sqrt{x} + 4}$ $\to$ $\infty$ as $x$ $\to$ $4$ from the right, then $x = 4$ is a vertical asymptote.

$x = 4$

Step 3

Evaluate $lim_{x \to \infty} \frac{\sqrt{x}}{x - 4 \sqrt{x} + 4}$ to find the horizontal asymptote.

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Step 3.1

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

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

Step 3.2

Evaluate the limit.

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Step 3.2.1

Simplify each term.

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Cancel the common factors.

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Step 3.2.2.3.1

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

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

Step 3.2.2.3.2

Cancel the common factor.

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

Step 3.2.2.3.3

Rewrite the expression.

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

Step 3.2.3

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

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

Step 3.2.4

Move the limit under the radical sign.

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

Step 3.3

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

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

Step 3.4

Evaluate the limit.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.5

Apply L'Hospital's rule.

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Step 3.5.1

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

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Step 3.5.1.1

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

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

Step 3.5.1.2

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

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

Step 3.5.1.3

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

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

Step 3.5.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.5.2

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 3.5.3

Find the derivative of the numerator and denominator.

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Step 3.5.3.1

Differentiate the numerator and denominator.

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{x}]}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.2

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

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{d}{d x} [x^{\frac{1}{2}}]}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.3

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

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} - 1}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.4

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

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

Step 3.5.3.5

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

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

Step 3.5.3.6

Combine the numerators over the common denominator.

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

Step 3.5.3.7

Simplify the numerator.

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Step 3.5.3.7.1

Multiply $- 1$ by $2$ .

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1 - 2}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.7.2

Subtract $2$ from $1$ .

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{- 1}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.8

Move the negative in front of the fraction.

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{1}{2} x^{- \frac{1}{2}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.9

Simplify.

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Step 3.5.3.9.1

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

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

Step 3.5.3.9.2

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

$\frac{\sqrt{0}}{1 - 4 lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x]} + lim_{x \to \infty} \frac{4}{x}}$

Step 3.5.3.10

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

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

Step 3.5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5.5

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

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

Step 3.5.6

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.9

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

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

Step 3.10

Simplify the answer.

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Step 3.10.1

Simplify the numerator.

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Step 3.10.1.1

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

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

Step 3.10.1.2

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

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

Step 3.10.2

Simplify the denominator.

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Step 3.10.2.1

Cancel the common factor of $2$ .

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Step 3.10.2.1.1

Factor $2$ out of $- 4$ .

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

Step 3.10.2.1.2

Cancel the common factor.

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

Step 3.10.2.1.3

Rewrite the expression.

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

Step 3.10.2.2

Multiply $- 2$ by $0$ .

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

Step 3.10.2.3

Multiply $4$ by $0$ .

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

Step 3.10.2.4

Add $1 + 0$ and $0$ .

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

Step 3.10.2.5

Add $1$ and $0$ .

$\frac{0}{1}$

Step 3.10.3

Divide $0$ by $1$ .

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

Step 5

Use polynomial division to find the oblique asymptotes. Because this expression contains a radical, polynomial division cannot be performed.

Cannot Find Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 4$

Horizontal Asymptotes: $y = 0$

Cannot Find Oblique Asymptotes

Step 7