Precalculus Examples

$f (x) = \frac{2 x^{2} + 3 x + 7}{\sqrt{x^{6} + 2}}$

Step 1

Find where the expression $\frac{2 x^{2} + 3 x + 7}{\sqrt{x^{6} + 2}}$ 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 $lim_{x \to \infty} \frac{2 x^{2} + 3 x + 7}{\sqrt{x^{6} + 2}}$ 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 $x^{3} = \sqrt{x^{6}}$ .

$lim_{x \to \infty} \frac{\frac{2 x^{2}}{x^{3}} + \frac{3 x}{x^{3}} + \frac{7}{x^{3}}}{\sqrt{\frac{x^{6}}{x^{6}} + \frac{2}{x^{6}}}}$

Step 3.2

Evaluate the limit.

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Step 3.2.1

Simplify each term.

$lim_{x \to \infty} \frac{\frac{2}{x} + \frac{3}{x^{2}} + \frac{7}{x^{3}}}{\sqrt{\frac{x^{6}}{x^{6}} + \frac{2}{x^{6}}}}$

Step 3.2.2

Cancel the common factor of $x^{6}$ .

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Step 3.2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{2}{x} + \frac{3}{x^{2}} + \frac{7}{x^{3}}}{\sqrt{\frac{x^{6}}{x^{6}} + \frac{2}{x^{6}}}}$

Step 3.2.2.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{2}{x} + \frac{3}{x^{2}} + \frac{7}{x^{3}}}{\sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.2.3

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

$\frac{lim_{x \to \infty} \frac{2}{x} + \frac{3}{x^{2}} + \frac{7}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.2.4

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

$\frac{lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{3}{x^{2}} + lim_{x \to \infty} \frac{7}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.2.5

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

$\frac{2 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{3}{x^{2}} + lim_{x \to \infty} \frac{7}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.3

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

$\frac{2 \cdot 0 + lim_{x \to \infty} \frac{3}{x^{2}} + lim_{x \to \infty} \frac{7}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \cdot 0 + 3 lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} \frac{7}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.5

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

$\frac{2 \cdot 0 + 3 \cdot 0 + lim_{x \to \infty} \frac{7}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.6

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

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.7

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

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 \cdot 0}{lim_{x \to \infty} \sqrt{1 + \frac{2}{x^{6}}}}$

Step 3.8

Evaluate the limit.

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Step 3.8.1

Move the limit under the radical sign.

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 \cdot 0}{\sqrt{lim_{x \to \infty} 1 + \frac{2}{x^{6}}}}$

Step 3.8.2

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

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 \cdot 0}{\sqrt{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{2}{x^{6}}}}$

Step 3.8.3

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

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 \cdot 0}{\sqrt{1 + lim_{x \to \infty} \frac{2}{x^{6}}}}$

Step 3.8.4

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

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 \cdot 0}{\sqrt{1 + 2 lim_{x \to \infty} \frac{1}{x^{6}}}}$

Step 3.9

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{6}}$ approaches $0$ .

$\frac{2 \cdot 0 + 3 \cdot 0 + 7 \cdot 0}{\sqrt{1 + 2 \cdot 0}}$

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

Multiply $2$ by $0$ .

$\frac{0 + 3 \cdot 0 + 7 \cdot 0}{\sqrt{1 + 2 \cdot 0}}$

Step 3.10.1.2

Multiply $3$ by $0$ .

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

Step 3.10.1.3

Multiply $7$ by $0$ .

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

Step 3.10.1.4

Add $0 + 0$ and $0$ .

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

Step 3.10.1.5

Add $0$ and $0$ .

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

Step 3.10.2

Simplify the denominator.

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Step 3.10.2.1

Multiply $2$ by $0$ .

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

Step 3.10.2.2

Add $1$ and $0$ .

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

Step 3.10.2.3

Any root of $1$ is $1$ .

$\frac{0}{1}$

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.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 0$

Cannot Find Oblique Asymptotes

Step 7