Calculus Examples

$lim_{x \to - \infty} \frac{\sqrt{x^{8} - 5 x^{3}}}{3 x^{4} + 4}$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Factor $x^{3}$ out of $x^{8} - 5 x^{3}$ .

Tap for more steps...

Step 1.1.1

Factor $x^{3}$ out of $x^{8}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{3} x^{5} - 5 x^{3}}}{3 x^{4} + 4}$

Step 1.1.2

Factor $x^{3}$ out of $- 5 x^{3}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{3} x^{5} + x^{3} \cdot - 5}}{3 x^{4} + 4}$

Step 1.1.3

Factor $x^{3}$ out of $x^{3} x^{5} + x^{3} \cdot - 5$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{3} (x^{5} - 5)}}{3 x^{4} + 4}$

Step 1.2

Rewrite $x^{3} (x^{5} - 5)$ as $x^{2} (x (x^{5} - 5))$ .

Tap for more steps...

Step 1.2.1

Factor out $x^{2}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} x (x^{5} - 5)}}{3 x^{4} + 4}$

Step 1.2.2

Add parentheses.

$lim_{x \to - \infty} \frac{\sqrt{x^{2} (x (x^{5} - 5))}}{3 x^{4} + 4}$

Step 1.3

Pull terms out from under the radical.

$lim_{x \to - \infty} \frac{x \sqrt{x (x^{5} - 5)}}{3 x^{4} + 4}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{4}$ .

$lim_{x \to - \infty} \frac{\frac{x \sqrt{x (x^{5} - 5)}}{x^{4}}}{\frac{3 x^{4}}{x^{4}} + \frac{4}{x^{4}}}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Cancel the common factor of $x$ and $x^{4}$ .

Tap for more steps...

Step 3.1.1

Factor $x$ out of $x \sqrt{x (x^{5} - 5)}$ .

$lim_{x \to - \infty} \frac{\frac{x (\sqrt{x (x^{5} - 5)})}{x^{4}}}{\frac{3 x^{4}}{x^{4}} + \frac{4}{x^{4}}}$

Step 3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1

Factor $x$ out of $x^{4}$ .

$lim_{x \to - \infty} \frac{\frac{x (\sqrt{x (x^{5} - 5)})}{x \cdot x^{3}}}{\frac{3 x^{4}}{x^{4}} + \frac{4}{x^{4}}}$

Step 3.1.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{x \sqrt{x (x^{5} - 5)}}{x \cdot x^{3}}}{\frac{3 x^{4}}{x^{4}} + \frac{4}{x^{4}}}$

Step 3.1.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{\sqrt{x (x^{5} - 5)}}{x^{3}}}{\frac{3 x^{4}}{x^{4}} + \frac{4}{x^{4}}}$

Step 3.2

Cancel the common factor of $x^{4}$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{x (x^{5} - 5)}}{x^{3}}}{3 + \frac{4}{x^{4}}}$

Step 3.3

Apply the distributive property.

$lim_{x \to - \infty} \frac{\frac{\sqrt{x \cdot x^{5} + x \cdot - 5}}{x^{3}}}{3 + \frac{4}{x^{4}}}$

Step 4

Multiply $x$ by $x^{5}$ by adding the exponents.

Tap for more steps...

Step 4.1

Multiply $x$ by $x^{5}$ .

Tap for more steps...

Step 4.1.1

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{x^{1} x^{5} + x \cdot - 5}}{x^{3}}}{3 + \frac{4}{x^{4}}}$

Step 4.1.2

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{x^{1 + 5} + x \cdot - 5}}{x^{3}}}{3 + \frac{4}{x^{4}}}$

Step 4.2

Add $1$ and $5$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{x^{6} + x \cdot - 5}}{x^{3}}}{3 + \frac{4}{x^{4}}}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Move $- 5$ to the left of $x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{x^{6} - 5 x}}{x^{3}}}{3 + \frac{4}{x^{4}}}$

Step 5.2

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

$\frac{lim_{x \to - \infty} \frac{\sqrt{x^{6} - 5 x}}{x^{3}}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 5.3

Factor $x$ out of $x^{6} - 5 x$ .

Tap for more steps...

Step 5.3.1

Factor $x$ out of $x^{6}$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{x \cdot x^{5} - 5 x}}{x^{3}}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 5.3.2

Factor $x$ out of $- 5 x$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{x \cdot x^{5} + x \cdot - 5}}{x^{3}}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 5.3.3

Factor $x$ out of $x \cdot x^{5} + x \cdot - 5$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{x (x^{5} - 5)}}{x^{3}}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 6

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $- \sqrt{x^{6}} = x^{3}$ .

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (x^{5} - 5)}{x^{6}}}}{\frac{x^{3}}{x^{3}}}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 7

Cancel the common factor of $x^{3}$ .

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (x^{5} - 5)}{x^{6}}}}{1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 8

Cancel the common factors.

Tap for more steps...

Step 8.1

Factor $x$ out of $x^{6}$ .

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (x^{5} - 5)}{x \cdot x^{5}}}}{1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 8.2

Cancel the common factor.

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (x^{5} - 5)}{x \cdot x^{5}}}}{1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 8.3

Rewrite the expression.

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{x^{5} - 5}{x^{5}}}}{1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 9

Evaluate the limit.

Tap for more steps...

Step 9.1

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

$\frac{\frac{lim_{x \to - \infty} - \sqrt{\frac{x^{5} - 5}{x^{5}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 9.2

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

$\frac{\frac{- lim_{x \to - \infty} \sqrt{\frac{x^{5} - 5}{x^{5}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 9.3

Move the limit under the radical sign.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{x^{5} - 5}{x^{5}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 10

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{5}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{x^{5}}{x^{5}} + \frac{- 5}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11

Evaluate the limit.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Cancel the common factor of $x^{5}$ .

Tap for more steps...

Step 11.1.1.1

Cancel the common factor.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{x^{5}}{x^{5}} + \frac{- 5}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.1.1.2

Rewrite the expression.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 + \frac{- 5}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.1.2

Move the negative in front of the fraction.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 - \frac{5}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.2

Cancel the common factor of $x^{5}$ .

Tap for more steps...

Step 11.2.1

Cancel the common factor.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 - \frac{5}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.2.2

Rewrite the expression.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 - \frac{5}{x^{5}}}{1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.3

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 1 - \frac{5}{x^{5}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.4

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{5}{x^{5}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.5

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

$\frac{\frac{- \sqrt{\frac{1 - lim_{x \to - \infty} \frac{5}{x^{5}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 11.6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{- \sqrt{\frac{1 - 5 lim_{x \to - \infty} \frac{1}{x^{5}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 12

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{5}}$ approaches $0$ .

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 13

Evaluate the limit.

Tap for more steps...

Step 13.1

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

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 13.2

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

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{1}}}{1}}{lim_{x \to - \infty} 3 + \frac{4}{x^{4}}}$

Step 13.3

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

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{1}}}{1}}{lim_{x \to - \infty} 3 + lim_{x \to - \infty} \frac{4}{x^{4}}}$

Step 13.4

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

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{1}}}{1}}{3 + lim_{x \to - \infty} \frac{4}{x^{4}}}$

Step 13.5

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

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{1}}}{1}}{3 + 4 lim_{x \to - \infty} \frac{1}{x^{4}}}$

Step 14

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

$\frac{\frac{- \sqrt{\frac{1 - 5 \cdot 0}{1}}}{1}}{3 + 4 \cdot 0}$

Step 15

Simplify the answer.

Tap for more steps...

Step 15.1

Divide $1 - 5 \cdot 0$ by $1$ .

$\frac{\frac{- \sqrt{1 - 5 \cdot 0}}{1}}{3 + 4 \cdot 0}$

Step 15.2

Divide $- \sqrt{1 - 5 \cdot 0}$ by $1$ .

$\frac{- \sqrt{1 - 5 \cdot 0}}{3 + 4 \cdot 0}$

Step 15.3

Simplify the numerator.

Tap for more steps...

Step 15.3.1

Multiply $- 5$ by $0$ .

$\frac{- \sqrt{1 + 0}}{3 + 4 \cdot 0}$

Step 15.3.2

Add $1$ and $0$ .

$\frac{- \sqrt{1}}{3 + 4 \cdot 0}$

Step 15.3.3

Any root of $1$ is $1$ .

$\frac{- 1 \cdot 1}{3 + 4 \cdot 0}$

Step 15.4

Simplify the denominator.

Tap for more steps...

Step 15.4.1

Multiply $4$ by $0$ .

$\frac{- 1 \cdot 1}{3 + 0}$

Step 15.4.2

Add $3$ and $0$ .

$\frac{- 1 \cdot 1}{3}$

Step 15.5

Multiply $- 1$ by $1$ .

$\frac{- 1}{3}$

Step 15.6

Move the negative in front of the fraction.

$- \frac{1}{3}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$