Calculus Examples

$lim_{x \to - \infty} \frac{x^{2} - 9}{x (x - 3)}$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$lim_{x \to - \infty} \frac{x^{2} - 9}{x \cdot x + x \cdot - 3}$

Step 1.2

Multiply $x$ by $x$ .

$lim_{x \to - \infty} \frac{x^{2} - 9}{x^{2} + x \cdot - 3}$

Step 1.3

Move $- 3$ to the left of $x$ .

$lim_{x \to - \infty} \frac{x^{2} - 9}{x^{2} - 3 x}$

Step 2

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

$lim_{x \to - \infty} \frac{\frac{x^{2}}{x^{2}} + \frac{- 9}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 3 x}{x^{2}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{x^{2}}{x^{2}} + \frac{- 9}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 3 x}{x^{2}}}$

Step 3.1.1.2

Rewrite the expression.

$lim_{x \to - \infty} \frac{1 + \frac{- 9}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 3 x}{x^{2}}}$

Step 3.1.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 3 x}{x^{2}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 3 x}{x^{2}}}$

Step 3.2.1.2

Rewrite the expression.

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{1 + \frac{- 3 x}{x^{2}}}$

Step 3.2.2

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

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Step 3.2.2.1

Factor $x$ out of $- 3 x$ .

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{1 + \frac{x \cdot - 3}{x^{2}}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

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

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{1 + \frac{x \cdot - 3}{x \cdot x}}$

Step 3.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{1 + \frac{x \cdot - 3}{x \cdot x}}$

Step 3.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{1 + \frac{- 3}{x}}$

Step 3.2.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{1 - \frac{9}{x^{2}}}{1 - \frac{3}{x}}$

Step 3.3

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

$\frac{lim_{x \to - \infty} 1 - \frac{9}{x^{2}}}{lim_{x \to - \infty} 1 - \frac{3}{x}}$

Step 3.4

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

$\frac{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{9}{x^{2}}}{lim_{x \to - \infty} 1 - \frac{3}{x}}$

Step 3.5

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

$\frac{1 - lim_{x \to - \infty} \frac{9}{x^{2}}}{lim_{x \to - \infty} 1 - \frac{3}{x}}$

Step 3.6

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - 9 lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 1 - \frac{3}{x}}$

Step 4

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

$\frac{1 - 9 \cdot 0}{lim_{x \to - \infty} 1 - \frac{3}{x}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{1 - 9 \cdot 0}{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{3}{x}}$

Step 5.2

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

$\frac{1 - 9 \cdot 0}{1 - lim_{x \to - \infty} \frac{3}{x}}$

Step 5.3

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

$\frac{1 - 9 \cdot 0}{1 - 3 lim_{x \to - \infty} \frac{1}{x}}$

Step 6

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

$\frac{1 - 9 \cdot 0}{1 - 3 \cdot 0}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- 9$ by $0$ .

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

Step 7.1.2

Add $1$ and $0$ .

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

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $- 3$ by $0$ .

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

Step 7.2.2

Add $1$ and $0$ .

$\frac{1}{1}$

Step 7.3

Divide $1$ by $1$ .

$1$