Calculus Examples

$lim_{x \to - \infty} \frac{7 - 3 x^{3}}{2 x^{3} + 1}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{7}{x^{3}} + \frac{- 3 x^{3}}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{1}{x^{3}}}$

Step 2

Evaluate the limit.

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Step 2.1

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

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Step 2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{7}{x^{3}} + \frac{- 3 x^{3}}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{1}{x^{3}}}$

Step 2.1.2

Divide $- 3$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{7}{x^{3}} - 3}{\frac{2 x^{3}}{x^{3}} + \frac{1}{x^{3}}}$

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{7}{x^{3}} - 3}{\frac{2 x^{3}}{x^{3}} + \frac{1}{x^{3}}}$

Step 2.2.2

Divide $2$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{7}{x^{3}} - 3}{2 + \frac{1}{x^{3}}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} \frac{7}{x^{3}} - 3}{lim_{x \to - \infty} 2 + \frac{1}{x^{3}}}$

Step 2.4

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

$\frac{lim_{x \to - \infty} \frac{7}{x^{3}} - lim_{x \to - \infty} 3}{lim_{x \to - \infty} 2 + \frac{1}{x^{3}}}$

Step 2.5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 lim_{x \to - \infty} \frac{1}{x^{3}} - lim_{x \to - \infty} 3}{lim_{x \to - \infty} 2 + \frac{1}{x^{3}}}$

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{7 \cdot 0 - lim_{x \to - \infty} 3}{lim_{x \to - \infty} 2 + \frac{1}{x^{3}}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{7 \cdot 0 - 1 \cdot 3}{lim_{x \to - \infty} 2 + \frac{1}{x^{3}}}$

Step 4.2

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

$\frac{7 \cdot 0 - 1 \cdot 3}{lim_{x \to - \infty} 2 + lim_{x \to - \infty} \frac{1}{x^{3}}}$

Step 4.3

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

$\frac{7 \cdot 0 - 1 \cdot 3}{2 + lim_{x \to - \infty} \frac{1}{x^{3}}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{7 \cdot 0 - 1 \cdot 3}{2 + 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $7$ by $0$ .

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

Step 6.1.2

Multiply $- 1$ by $3$ .

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

Step 6.1.3

Subtract $3$ from $0$ .

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

Step 6.2

Add $2$ and $0$ .

$\frac{- 3}{2}$

Step 6.3

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$