Using the limit laws, find limx → a e(x).

By the quotient rule for limits, since limx→ a g(x) = 3 ≠ 0, we have limx→ a f(x)/g(x) = fraclimx→ a f(x)limx→ a g(x) = 6/3 = 2. Now apply the power rule: since e(x) = [f(x)/g(x)]^2, limx→ a e(x) = left(limx→ a f(x)/g(x)right)^2 = 2^2 = 4.

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Combining quotient and power rules

Suppose $\lim_{x \to a} f(x) = 6$ and $\lim_{x \to a} g(x) = 3$ . Define $e(x) = \left[\dfrac{f(x)}{g(x)}\right]^2$ .

Using the limit laws, find $\lim_{x \to a} e(x)$ .

Hint

First apply the quotient rule to find $\lim_{x\to a} f(x)/g(x)$ , then apply the power rule to square that result.

Step-by-step solution

By the quotient rule for limits, since $\lim_{x\to a} g(x) = 3 \neq 0$ , we have $\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)} = \frac{6}{3} = 2.$

Now apply the power rule: since $e(x) = [f(x)/g(x)]^2$ , $\lim_{x\to a} e(x) = \left(\lim_{x\to a} \frac{f(x)}{g(x)}\right)^2 = 2^2 = 4.$

Combining quotient and power rules

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