Calculus Examples

$lim_{x \to - \infty} \frac{- 3 x^{3} + 4 x^{2}}{4 x + 6 x^{3}}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{- 3 x^{3}}{x^{3}} + \frac{4 x^{2}}{x^{3}}}{\frac{4 x}{x^{3}} + \frac{6 x^{3}}{x^{3}}}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{- 3 x^{3}}{x^{3}} + \frac{4 x^{2}}{x^{3}}}{\frac{4 x}{x^{3}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.1.1.2

Divide $- 3$ by $1$ .

$lim_{x \to - \infty} \frac{- 3 + \frac{4 x^{2}}{x^{3}}}{\frac{4 x}{x^{3}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.1.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

Tap for more steps...

Step 2.1.2.1

Factor $x^{2}$ out of $4 x^{2}$ .

$lim_{x \to - \infty} \frac{- 3 + \frac{x^{2} \cdot 4}{x^{3}}}{\frac{4 x}{x^{3}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.1.2.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to - \infty} \frac{- 3 + \frac{x^{2} \cdot 4}{x^{2} x}}{\frac{4 x}{x^{3}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 3 + \frac{x^{2} \cdot 4}{x^{2} x}}{\frac{4 x}{x^{3}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

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

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Factor $x$ out of $4 x$ .

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

Step 2.2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1.2.1

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

$lim_{x \to - \infty} \frac{- 3 + \frac{4}{x}}{\frac{x \cdot 4}{x \cdot x^{2}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.2.1.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 3 + \frac{4}{x}}{\frac{x \cdot 4}{x \cdot x^{2}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.2.1.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 3 + \frac{4}{x}}{\frac{4}{x^{2}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 3 + \frac{4}{x}}{\frac{4}{x^{2}} + \frac{6 x^{3}}{x^{3}}}$

Step 2.2.2.2

Divide $6$ by $1$ .

$lim_{x \to - \infty} \frac{- 3 + \frac{4}{x}}{\frac{4}{x^{2}} + 6}$

Step 2.3

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

$\frac{lim_{x \to - \infty} - 3 + \frac{4}{x}}{lim_{x \to - \infty} \frac{4}{x^{2}} + 6}$

Step 2.4

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

$\frac{- lim_{x \to - \infty} 3 + lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} \frac{4}{x^{2}} + 6}$

Step 2.5

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

$\frac{- 1 \cdot 3 + lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} \frac{4}{x^{2}} + 6}$

Step 2.6

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

$\frac{- 3 + 4 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} \frac{4}{x^{2}} + 6}$

Step 3

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

$\frac{- 3 + 4 \cdot 0}{lim_{x \to - \infty} \frac{4}{x^{2}} + 6}$

Step 4

Evaluate the limit.

Tap for more steps...

Step 4.1

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

$\frac{- 3 + 4 \cdot 0}{lim_{x \to - \infty} \frac{4}{x^{2}} + lim_{x \to - \infty} 6}$

Step 4.2

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

$\frac{- 3 + 4 \cdot 0}{4 lim_{x \to - \infty} \frac{1}{x^{2}} + lim_{x \to - \infty} 6}$

Step 5

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

$\frac{- 3 + 4 \cdot 0}{4 \cdot 0 + lim_{x \to - \infty} 6}$

Step 6

Evaluate the limit.

Tap for more steps...

Step 6.1

Evaluate the limit of $6$ which is constant as $x$ approaches $- \infty$ .

$\frac{- 3 + 4 \cdot 0}{4 \cdot 0 + 6}$

Step 6.2

Simplify the answer.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $- 3 + 4 \cdot 0$ and $4 \cdot 0 + 6$ .

Tap for more steps...

Step 6.2.1.1

Reorder terms.

$\frac{- 3 + 0 \cdot 4}{4 \cdot 0 + 6}$

Step 6.2.1.2

Factor $3$ out of $- 3$ .

$\frac{3 (- 1) + 0 \cdot 4}{4 \cdot 0 + 6}$

Step 6.2.1.3

Factor $3$ out of $0 \cdot 4$ .

$\frac{3 (- 1) + 3 (0 \cdot 4)}{4 \cdot 0 + 6}$

Step 6.2.1.4

Factor $3$ out of $3 (- 1) + 3 (0 \cdot 4)$ .

$\frac{3 (- 1 + 0 \cdot 4)}{4 \cdot 0 + 6}$

Step 6.2.1.5

Cancel the common factors.

Tap for more steps...

Step 6.2.1.5.1

Factor $3$ out of $4 \cdot 0$ .

$\frac{3 (- 1 + 0 \cdot 4)}{3 (4 \cdot 0) + 6}$

Step 6.2.1.5.2

Factor $3$ out of $6$ .

$\frac{3 (- 1 + 0 \cdot 4)}{3 (4 \cdot 0) + 3 \cdot 2}$

Step 6.2.1.5.3

Factor $3$ out of $3 (4 \cdot 0) + 3 (2)$ .

$\frac{3 (- 1 + 0 \cdot 4)}{3 (4 \cdot 0 + 2)}$

Step 6.2.1.5.4

Cancel the common factor.

$\frac{3 (- 1 + 0 \cdot 4)}{3 (4 \cdot 0 + 2)}$

Step 6.2.1.5.5

Rewrite the expression.

$\frac{- 1 + 0 \cdot 4}{4 \cdot 0 + 2}$

Step 6.2.2

Simplify the numerator.

Tap for more steps...

Step 6.2.2.1

Multiply $0$ by $4$ .

$\frac{- 1 + 0}{4 \cdot 0 + 2}$

Step 6.2.2.2

Add $- 1$ and $0$ .

$\frac{- 1}{4 \cdot 0 + 2}$

Step 6.2.3

Simplify the denominator.

Tap for more steps...

Step 6.2.3.1

Multiply $4$ by $0$ .

$\frac{- 1}{0 + 2}$

Step 6.2.3.2

Add $0$ and $2$ .

$\frac{- 1}{2}$

Step 6.2.4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$