Calculus Examples

$lim_{x \to - \infty} \frac{6}{x} - \frac{x}{9}$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{x \to - \infty} \frac{6}{x} - lim_{x \to - \infty} \frac{x}{9}$

Step 1.2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} \frac{x}{9}$

Step 2

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

$6 \cdot 0 - lim_{x \to - \infty} \frac{x}{9}$

Step 3

Evaluate the limit.

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Step 3.1

The limit at negative infinity of a polynomial of odd degree whose leading coefficient is positive is negative infinity.

$6 \cdot 0 - - \infty$

Step 3.2

Simplify the answer.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $6$ by $0$ .

$0 - - \infty$

Step 3.2.1.2

A non-zero constant times infinity is infinity.

$0 + \infty$

Step 3.2.2

Add $0$ and $\infty$ .

$\infty$