Calculus Examples

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

Step 1

Move the term $\frac{1}{- 3}$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

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

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

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

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

Tap for more steps...

Step 3.1.3.1

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

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

Step 3.1.3.2

Factor $x$ out of $x^{1}$ .

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

Step 3.1.3.3

Cancel the common factors.

Tap for more steps...

Step 3.1.3.3.1

Factor $x$ out of $x^{2}$ .

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

Step 3.1.3.3.2

Cancel the common factor.

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

Step 3.1.3.3.3

Rewrite the expression.

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$\frac{1}{- 3} lim_{x \to \infty} \frac{- 1 - \frac{1}{x}}{1}$

Step 3.3

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

$\frac{1}{- 3} \cdot \frac{lim_{x \to \infty} - 1 - \frac{1}{x}}{lim_{x \to \infty} 1}$

Step 3.4

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

$\frac{1}{- 3} \cdot \frac{- lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}$

Step 3.5

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

$\frac{1}{- 3} \cdot \frac{- 1 \cdot 1 - lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}$

Step 4

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

$\frac{1}{- 3} \cdot \frac{- 1 - 0}{lim_{x \to \infty} 1}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify the answer.

Tap for more steps...

Step 5.2.1

Move the negative in front of the fraction.

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

Step 5.2.2

Divide $- 1 - 0$ by $1$ .

$- \frac{1}{3} (- 1 - 0)$

Step 5.2.3

Subtract $0$ from $- 1$ .

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

Step 5.2.4

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

Tap for more steps...

Step 5.2.4.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{3})$

Step 5.2.4.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$