← Back to learning objective

Product rule for limits

Suppose f(x)f(x) and g(x)g(x) are functions defined near x=ax=a such that lim⁡x→af(x)=4\lim_{x\to a} f(x) = 4 and lim⁡x→ag(x)=−2\lim_{x\to a} g(x) = -2.

Let q(x)=f(x)⋅g(x)q(x) = f(x)\cdot g(x). Using the limit laws, find lim⁡x→aq(x)\lim_{x\to a} q(x).

Hint

Apply the product rule for limits: the limit of a product equals the product of the limits, provided each individual limit exists.

Step-by-step solution

Since lim⁡x→af(x)=4\lim_{x\to a} f(x) = 4 and lim⁡x→ag(x)=−2\lim_{x\to a} g(x) = -2 both exist, the product rule for limits applies: lim⁡x→a[f(x)g(x)]=(lim⁡x→af(x))(lim⁡x→ag(x))\lim_{x\to a}[f(x)g(x)] = \left(\lim_{x\to a} f(x)\right)\left(\lim_{x\to a} g(x)\right).

Substituting the known values gives lim⁡x→aq(x)=(4)(−2)=−8\lim_{x\to a} q(x) = (4)(-2) = -8.

Quotient rule for limits

Using the same two limits, is the limit of f(x)g(x)\frac{f(x)}{g(x)} positive or negative?

Hint
Step-by-step solution

Since lim⁡x→af(x)=4\lim_{x\to a} f(x) = 4 and lim⁡x→ag(x)=−2\lim_{x\to a} g(x) = -2 both exist, and the limit of gg is not zero, the quotient rule for limits applies: lim⁡x→af(x)g(x)=lim⁡x→af(x)lim⁡x→ag(x)\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)}.

Substituting the known values gives lim⁡x→af(x)g(x)=4−2=−2\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{4}{-2} = -2. Since this value is negative, the limit is negative.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap