Calculus Examples

$f (x) = \frac{2 x}{\sqrt{x^{2} + 7}}$

Step 1

Find where the expression $\frac{2 x \sqrt{x^{2} + 7}}{x^{2} + 7}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} \frac{2 x \sqrt{x^{2} + 7}}{x^{2} + 7}$ to find the horizontal asymptote.

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Step 3.1

Reduce.

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 7}$ as ${(x^{2} + 7)}^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{2 x {(x^{2} + 7)}^{\frac{1}{2}}}{x^{2} + 7}$

Step 3.1.2

Factor $x^{2} + 7$ out of $2 x {(x^{2} + 7)}^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{(x^{2} + 7) (2 x {(x^{2} + 7)}^{- \frac{1}{2}})}{x^{2} + 7}$

Step 3.1.3

Cancel the common factors.

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Step 3.1.3.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{(x^{2} + 7) (2 x {(x^{2} + 7)}^{- \frac{1}{2}})}{(x^{2} + 7) \cdot 1}$

Step 3.1.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{(x^{2} + 7) (2 x {(x^{2} + 7)}^{- \frac{1}{2}})}{(x^{2} + 7) \cdot 1}$

Step 3.1.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{2 x {(x^{2} + 7)}^{- \frac{1}{2}}}{1}$

Step 3.1.3.4

Divide $2 x {(x^{2} + 7)}^{- \frac{1}{2}}$ by $1$ .

$lim_{x \to \infty} 2 x {(x^{2} + 7)}^{- \frac{1}{2}}$

Step 3.1.4

Move ${(x^{2} + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to \infty} \frac{2 x}{{(x^{2} + 7)}^{\frac{1}{2}}}$

Step 3.2

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

$2 lim_{x \to \infty} \frac{x}{{(x^{2} + 7)}^{\frac{1}{2}}}$

Step 3.3

Rewrite ${(x^{2} + 7)}^{\frac{1}{2}}$ as $\sqrt{x^{2} + 7}$ .

$2 lim_{x \to \infty} \frac{x}{\sqrt{x^{2} + 7}}$

Step 3.4

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

$2 lim_{x \to \infty} \frac{\frac{x}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{7}{x^{2}}}}$

Step 3.5

Evaluate the limit.

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Step 3.5.1

Cancel the common factor of $x$ .

$2 lim_{x \to \infty} \frac{1}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{7}{x^{2}}}}$

Step 3.5.2

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

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Step 3.5.2.1

Cancel the common factor.

$2 lim_{x \to \infty} \frac{1}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{7}{x^{2}}}}$

Step 3.5.2.2

Rewrite the expression.

$2 lim_{x \to \infty} \frac{1}{\sqrt{1 + \frac{7}{x^{2}}}}$

Step 3.5.3

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

$2 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \sqrt{1 + \frac{7}{x^{2}}}}$

Step 3.5.4

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

$2 \frac{1}{lim_{x \to \infty} \sqrt{1 + \frac{7}{x^{2}}}}$

Step 3.5.5

Move the limit under the radical sign.

$2 \frac{1}{\sqrt{lim_{x \to \infty} 1 + \frac{7}{x^{2}}}}$

Step 3.5.6

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

$2 \frac{1}{\sqrt{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{7}{x^{2}}}}$

Step 3.5.7

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

$2 \frac{1}{\sqrt{1 + lim_{x \to \infty} \frac{7}{x^{2}}}}$

Step 3.5.8

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{1}{\sqrt{1 + 7 lim_{x \to \infty} \frac{1}{x^{2}}}}$

Step 3.6

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

$2 \frac{1}{\sqrt{1 + 7 \cdot 0}}$

Step 3.7

Simplify the answer.

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Step 3.7.1

Simplify the denominator.

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Step 3.7.1.1

Multiply $7$ by $0$ .

$2 \frac{1}{\sqrt{1 + 0}}$

Step 3.7.1.2

Add $1$ and $0$ .

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

Step 3.7.1.3

Any root of $1$ is $1$ .

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

Step 3.7.2

Cancel the common factor of $1$ .

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Step 3.7.2.1

Cancel the common factor.

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

Step 3.7.2.2

Rewrite the expression.

$2 \cdot 1$

Step 3.7.3

Multiply $2$ by $1$ .

$2$

Step 4

Evaluate $lim_{x \to - \infty} \frac{2 x \sqrt{x^{2} + 7}}{x^{2} + 7}$ to find the horizontal asymptote.

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Step 4.1

Reduce.

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 7}$ as ${(x^{2} + 7)}^{\frac{1}{2}}$ .

$lim_{x \to - \infty} \frac{2 x {(x^{2} + 7)}^{\frac{1}{2}}}{x^{2} + 7}$

Step 4.1.2

Factor $x^{2} + 7$ out of $2 x {(x^{2} + 7)}^{\frac{1}{2}}$ .

$lim_{x \to - \infty} \frac{(x^{2} + 7) (2 x {(x^{2} + 7)}^{- \frac{1}{2}})}{x^{2} + 7}$

Step 4.1.3

Cancel the common factors.

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Step 4.1.3.1

Multiply by $1$ .

$lim_{x \to - \infty} \frac{(x^{2} + 7) (2 x {(x^{2} + 7)}^{- \frac{1}{2}})}{(x^{2} + 7) \cdot 1}$

Step 4.1.3.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{(x^{2} + 7) (2 x {(x^{2} + 7)}^{- \frac{1}{2}})}{(x^{2} + 7) \cdot 1}$

Step 4.1.3.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{2 x {(x^{2} + 7)}^{- \frac{1}{2}}}{1}$

Step 4.1.3.4

Divide $2 x {(x^{2} + 7)}^{- \frac{1}{2}}$ by $1$ .

$lim_{x \to - \infty} 2 x {(x^{2} + 7)}^{- \frac{1}{2}}$

Step 4.1.4

Move ${(x^{2} + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to - \infty} \frac{2 x}{{(x^{2} + 7)}^{\frac{1}{2}}}$

Step 4.2

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

$2 lim_{x \to - \infty} \frac{x}{{(x^{2} + 7)}^{\frac{1}{2}}}$

Step 4.3

Rewrite ${(x^{2} + 7)}^{\frac{1}{2}}$ as $\sqrt{x^{2} + 7}$ .

$2 lim_{x \to - \infty} \frac{x}{\sqrt{x^{2} + 7}}$

Step 4.4

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

$2 lim_{x \to - \infty} \frac{\frac{x}{x}}{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{7}{x^{2}}}}$

Step 4.5

Evaluate the limit.

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Step 4.5.1

Cancel the common factor of $x$ .

$2 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{7}{x^{2}}}}$

Step 4.5.2

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

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Step 4.5.2.1

Cancel the common factor.

$2 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{7}{x^{2}}}}$

Step 4.5.2.2

Rewrite the expression.

$2 lim_{x \to - \infty} \frac{1}{- \sqrt{1 + \frac{7}{x^{2}}}}$

Step 4.5.3

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

$2 \frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} - \sqrt{1 + \frac{7}{x^{2}}}}$

Step 4.5.4

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

$2 \frac{1}{lim_{x \to - \infty} - \sqrt{1 + \frac{7}{x^{2}}}}$

Step 4.5.5

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{1}{- lim_{x \to - \infty} \sqrt{1 + \frac{7}{x^{2}}}}$

Step 4.5.6

Move the limit under the radical sign.

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} 1 + \frac{7}{x^{2}}}}$

Step 4.5.7

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

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{7}{x^{2}}}}$

Step 4.5.8

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

$2 \frac{1}{- \sqrt{1 + lim_{x \to - \infty} \frac{7}{x^{2}}}}$

Step 4.5.9

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{1}{- \sqrt{1 + 7 lim_{x \to - \infty} \frac{1}{x^{2}}}}$

Step 4.6

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

$2 \frac{1}{- \sqrt{1 + 7 \cdot 0}}$

Step 4.7

Simplify the answer.

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Step 4.7.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.7.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$2 \frac{- 1 (- 1)}{- \sqrt{1 + 7 \cdot 0}}$

Step 4.7.1.2

Move the negative in front of the fraction.

$2 (- \frac{1}{\sqrt{1 + 7 \cdot 0}})$

Step 4.7.2

Simplify the denominator.

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Step 4.7.2.1

Multiply $7$ by $0$ .

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

Step 4.7.2.2

Add $1$ and $0$ .

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

Step 4.7.2.3

Any root of $1$ is $1$ .

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

Step 4.7.3

Cancel the common factor of $1$ .

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Step 4.7.3.1

Cancel the common factor.

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

Step 4.7.3.2

Rewrite the expression.

$2 (- 1 \cdot 1)$

Step 4.7.4

Multiply $2 (- 1 \cdot 1)$ .

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Step 4.7.4.1

Multiply $- 1$ by $1$ .

$2 \cdot - 1$

Step 4.7.4.2

Multiply $2$ by $- 1$ .

$- 2$

Step 5

List the horizontal asymptotes:

$y = 2, - 2$

Step 6

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

Cannot Find Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 2, - 2$

Cannot Find Oblique Asymptotes

Step 8