Calculus Examples

$\frac{3 x - 2}{\sqrt{2 x^{2} + 1}}$

Step 1

Find where the expression $\frac{3 x - 2}{\sqrt{2 x^{2} + 1}}$ 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 $lim_{x \to \infty} \frac{3 x - 2}{\sqrt{2 x^{2} + 1}}$ to find the horizontal asymptote.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Evaluate the limit.

Tap for more steps...

Step 3.2.1

Simplify each term.

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

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Divide $2$ by $1$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

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

Step 3.2.6

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

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

Step 3.3

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

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

Step 3.4

Evaluate the limit.

Tap for more steps...

Step 3.4.1

Move the limit under the radical sign.

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

Step 3.4.2

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

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

Step 3.4.3

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

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

Step 3.5

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

$\frac{3 - 2 \cdot 0}{\sqrt{2 + 0}}$

Step 3.6

Simplify the answer.

Tap for more steps...

Step 3.6.1

Simplify the numerator.

Tap for more steps...

Step 3.6.1.1

Multiply $- 2$ by $0$ .

$\frac{3 + 0}{\sqrt{2 + 0}}$

Step 3.6.1.2

Add $3$ and $0$ .

$\frac{3}{\sqrt{2 + 0}}$

Step 3.6.2

Add $2$ and $0$ .

$\frac{3}{\sqrt{2}}$

Step 3.6.3

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.6.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.6.4.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.6.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.6.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.6.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.6.4.5

Add $1$ and $1$ .

$\frac{3 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.6.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.6.4.6.1

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

$\frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.6.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.6.4.6.3

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

$\frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.6.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.4.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.6.4.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{2}}{2^{1}}$

Step 3.6.4.6.5

Evaluate the exponent.

$\frac{3 \sqrt{2}}{2}$

Step 4

Evaluate $lim_{x \to - \infty} \frac{3 x - 2}{\sqrt{2 x^{2} + 1}}$ to find the horizontal asymptote.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Evaluate the limit.

Tap for more steps...

Step 4.2.1

Simplify each term.

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

Step 4.2.2

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

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Divide $2$ by $1$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Evaluate the limit of $3$ which is constant as $x$ approaches $- \infty$ .

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

Step 4.2.6

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

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

Step 4.3

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

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

Step 4.4

Evaluate the limit.

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

Move the limit under the radical sign.

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

Step 4.4.3

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

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

Step 4.4.4

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

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

Step 4.5

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

$\frac{3 - 2 \cdot 0}{- \sqrt{2 + 0}}$

Step 4.6

Simplify the answer.

Tap for more steps...

Step 4.6.1

Simplify the numerator.

Tap for more steps...

Step 4.6.1.1

Multiply $- 2$ by $0$ .

$\frac{3 + 0}{- \sqrt{2 + 0}}$

Step 4.6.1.2

Add $3$ and $0$ .

$\frac{3}{- \sqrt{2 + 0}}$

Step 4.6.2

Add $2$ and $0$ .

$\frac{3}{- \sqrt{2}}$

Step 4.6.3

Move the negative in front of the fraction.

$- \frac{3}{\sqrt{2}}$

Step 4.6.4

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- (\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 4.6.5

Combine and simplify the denominator.

Tap for more steps...

Step 4.6.5.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.6.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.6.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.6.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.6.5.5

Add $1$ and $1$ .

$- \frac{3 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.6.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 4.6.5.6.1

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

$- \frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.6.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.6.5.6.3

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

$- \frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.6.5.6.4.1

Cancel the common factor.

$- \frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.5.6.4.2

Rewrite the expression.

$- \frac{3 \sqrt{2}}{2^{1}}$

Step 4.6.5.6.5

Evaluate the exponent.

$- \frac{3 \sqrt{2}}{2}$

Step 5

List the horizontal asymptotes:

$y = \frac{3 \sqrt{2}}{2}, - \frac{3 \sqrt{2}}{2}$

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: $y = \frac{3 \sqrt{2}}{2}, - \frac{3 \sqrt{2}}{2}$

Cannot Find Oblique Asymptotes

Step 8