Calculus Examples

$f (x) = \frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$

Step 1

Find where the expression $\frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ is undefined.

$- \frac{\sqrt{3}}{3} \leq x \leq \frac{\sqrt{3}}{3}$

Step 2

Since $\frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ $\to$ $- \infty$ as $x$ $\to$ $- \frac{\sqrt{3}}{3}$ from the left and $\frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $- \frac{\sqrt{3}}{3}$ from the right, then $x = - \frac{\sqrt{3}}{3}$ is a vertical asymptote.

$x = - \frac{\sqrt{3}}{3}$

Step 3

Since $\frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $\frac{\sqrt{3}}{3}$ from the left and $\frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ $\to$ $- \infty$ as $x$ $\to$ $\frac{\sqrt{3}}{3}$ from the right, then $x = \frac{\sqrt{3}}{3}$ is a vertical asymptote.

$x = \frac{\sqrt{3}}{3}$

Step 4

List all of the vertical asymptotes:

$x = - \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}$

Step 5

Evaluate $lim_{x \to \infty} \frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ to find the horizontal asymptote.

Tap for more steps...

Step 5.1

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

$lim_{x \to \infty} \frac{\frac{4 x}{x} + \frac{- 3}{x}}{\sqrt{\frac{3 x^{2}}{x^{2}} + \frac{- 1}{x^{2}}}}$

Step 5.2

Evaluate the limit.

Tap for more steps...

Step 5.2.1

Simplify each term.

$lim_{x \to \infty} \frac{4 - \frac{3}{x}}{\sqrt{\frac{3 x^{2}}{x^{2}} + \frac{- 1}{x^{2}}}}$

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

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

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{4 - \frac{3}{x}}{\sqrt{\frac{3 x^{2}}{x^{2}} + \frac{- 1}{x^{2}}}}$

Step 5.2.2.1.2

Divide $3$ by $1$ .

$lim_{x \to \infty} \frac{4 - \frac{3}{x}}{\sqrt{3 + \frac{- 1}{x^{2}}}}$

Step 5.2.2.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{4 - \frac{3}{x}}{\sqrt{3 - \frac{1}{x^{2}}}}$

Step 5.2.3

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

$\frac{lim_{x \to \infty} 4 - \frac{3}{x}}{lim_{x \to \infty} \sqrt{3 - \frac{1}{x^{2}}}}$

Step 5.2.4

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

$\frac{lim_{x \to \infty} 4 - lim_{x \to \infty} \frac{3}{x}}{lim_{x \to \infty} \sqrt{3 - \frac{1}{x^{2}}}}$

Step 5.2.5

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{4 - lim_{x \to \infty} \frac{3}{x}}{lim_{x \to \infty} \sqrt{3 - \frac{1}{x^{2}}}}$

Step 5.2.6

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

$\frac{4 - 3 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt{3 - \frac{1}{x^{2}}}}$

Step 5.3

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

$\frac{4 - 3 \cdot 0}{lim_{x \to \infty} \sqrt{3 - \frac{1}{x^{2}}}}$

Step 5.4

Evaluate the limit.

Tap for more steps...

Step 5.4.1

Move the limit under the radical sign.

$\frac{4 - 3 \cdot 0}{\sqrt{lim_{x \to \infty} 3 - \frac{1}{x^{2}}}}$

Step 5.4.2

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

$\frac{4 - 3 \cdot 0}{\sqrt{lim_{x \to \infty} 3 - lim_{x \to \infty} \frac{1}{x^{2}}}}$

Step 5.4.3

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

$\frac{4 - 3 \cdot 0}{\sqrt{3 - lim_{x \to \infty} \frac{1}{x^{2}}}}$

Step 5.5

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

$\frac{4 - 3 \cdot 0}{\sqrt{3 - 0}}$

Step 5.6

Simplify the answer.

Tap for more steps...

Step 5.6.1

Simplify the numerator.

Tap for more steps...

Step 5.6.1.1

Multiply $- 3$ by $0$ .

$\frac{4 + 0}{\sqrt{3 - 0}}$

Step 5.6.1.2

Add $4$ and $0$ .

$\frac{4}{\sqrt{3 - 0}}$

Step 5.6.2

Simplify the denominator.

Tap for more steps...

Step 5.6.2.1

Multiply $- 1$ by $0$ .

$\frac{4}{\sqrt{3 + 0}}$

Step 5.6.2.2

Add $3$ and $0$ .

$\frac{4}{\sqrt{3}}$

Step 5.6.3

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.6.4

Combine and simplify the denominator.

Tap for more steps...

Step 5.6.4.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.6.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.6.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.6.4.4

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

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.6.4.5

Add $1$ and $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.6.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 5.6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.6.4.6.2

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

$\frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.6.4.6.3

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

$\frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.6.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.6.4.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.6.4.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{3}}{3^{1}}$

Step 5.6.4.6.5

Evaluate the exponent.

$\frac{4 \sqrt{3}}{3}$

Step 6

Evaluate $lim_{x \to - \infty} \frac{4 x - 3}{\sqrt{3 x^{2} - 1}}$ to find the horizontal asymptote.

Tap for more steps...

Step 6.1

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

$lim_{x \to - \infty} \frac{\frac{4 x}{x} + \frac{- 3}{x}}{- \sqrt{\frac{3 x^{2}}{x^{2}} + \frac{- 1}{x^{2}}}}$

Step 6.2

Evaluate the limit.

Tap for more steps...

Step 6.2.1

Simplify each term.

$lim_{x \to - \infty} \frac{4 - \frac{3}{x}}{- \sqrt{\frac{3 x^{2}}{x^{2}} + \frac{- 1}{x^{2}}}}$

Step 6.2.2

Simplify each term.

Tap for more steps...

Step 6.2.2.1

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

Tap for more steps...

Step 6.2.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{4 - \frac{3}{x}}{- \sqrt{\frac{3 x^{2}}{x^{2}} + \frac{- 1}{x^{2}}}}$

Step 6.2.2.1.2

Divide $3$ by $1$ .

$lim_{x \to - \infty} \frac{4 - \frac{3}{x}}{- \sqrt{3 + \frac{- 1}{x^{2}}}}$

Step 6.2.2.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{4 - \frac{3}{x}}{- \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.2.3

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

$\frac{lim_{x \to - \infty} 4 - \frac{3}{x}}{lim_{x \to - \infty} - \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.2.4

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

$\frac{lim_{x \to - \infty} 4 - lim_{x \to - \infty} \frac{3}{x}}{lim_{x \to - \infty} - \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.2.5

Evaluate the limit of $4$ which is constant as $x$ approaches $- \infty$ .

$\frac{4 - lim_{x \to - \infty} \frac{3}{x}}{lim_{x \to - \infty} - \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.2.6

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

$\frac{4 - 3 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} - \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.3

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

$\frac{4 - 3 \cdot 0}{lim_{x \to - \infty} - \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.4

Evaluate the limit.

Tap for more steps...

Step 6.4.1

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

$\frac{4 - 3 \cdot 0}{- lim_{x \to - \infty} \sqrt{3 - \frac{1}{x^{2}}}}$

Step 6.4.2

Move the limit under the radical sign.

$\frac{4 - 3 \cdot 0}{- \sqrt{lim_{x \to - \infty} 3 - \frac{1}{x^{2}}}}$

Step 6.4.3

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

$\frac{4 - 3 \cdot 0}{- \sqrt{lim_{x \to - \infty} 3 - lim_{x \to - \infty} \frac{1}{x^{2}}}}$

Step 6.4.4

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

$\frac{4 - 3 \cdot 0}{- \sqrt{3 - lim_{x \to - \infty} \frac{1}{x^{2}}}}$

Step 6.5

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

$\frac{4 - 3 \cdot 0}{- \sqrt{3 - 0}}$

Step 6.6

Simplify the answer.

Tap for more steps...

Step 6.6.1

Simplify the numerator.

Tap for more steps...

Step 6.6.1.1

Multiply $- 3$ by $0$ .

$\frac{4 + 0}{- \sqrt{3 - 0}}$

Step 6.6.1.2

Add $4$ and $0$ .

$\frac{4}{- \sqrt{3 - 0}}$

Step 6.6.2

Simplify the denominator.

Tap for more steps...

Step 6.6.2.1

Multiply $- 1$ by $0$ .

$\frac{4}{- \sqrt{3 + 0}}$

Step 6.6.2.2

Add $3$ and $0$ .

$\frac{4}{- \sqrt{3}}$

Step 6.6.3

Move the negative in front of the fraction.

$- \frac{4}{\sqrt{3}}$

Step 6.6.4

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- (\frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 6.6.5

Combine and simplify the denominator.

Tap for more steps...

Step 6.6.5.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.6.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 6.6.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 6.6.5.4

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

$- \frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 6.6.5.5

Add $1$ and $1$ .

$- \frac{4 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 6.6.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 6.6.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 6.6.5.6.2

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

$- \frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 6.6.5.6.3

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

$- \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.6.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.6.5.6.4.1

Cancel the common factor.

$- \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.6.5.6.4.2

Rewrite the expression.

$- \frac{4 \sqrt{3}}{3^{1}}$

Step 6.6.5.6.5

Evaluate the exponent.

$- \frac{4 \sqrt{3}}{3}$

Step 7

List the horizontal asymptotes:

$y = \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}$

Step 8

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

Cannot Find Oblique Asymptotes

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}$

Horizontal Asymptotes: $y = \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}$

Cannot Find Oblique Asymptotes

Step 10