Let z(x) = [f(x)]^1/2 · g(x). Using the limit laws, find limx → a z(x).

Since limx→ a f(x) = 9, the power rule gives limx→ a [f(x)]^1/2 = left(limx→ a f(x)right)^1/2 = 9^1/2 = 3. Now apply the product rule: limx→ a z(x) = limx→ a [f(x)]^1/2 · limx→ a g(x) = 3 · 5 = 15.

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Limit of a product involving a power rule

Suppose $\lim_{x \to a} f(x) = 9$ and $\lim_{x \to a} g(x) = 5$ .

Let $z(x) = [f(x)]^{1/2} \cdot g(x)$ . Using the limit laws, find $\lim_{x \to a} z(x).$

Hint

Apply the power rule to find $\lim_{x\to a}[f(x)]^{1/2}$ first, then use the product rule to combine it with $\lim_{x\to a} g(x)$ .

Step-by-step solution

Since $\lim_{x\to a} f(x) = 9$ , the power rule gives $\lim_{x\to a} [f(x)]^{1/2} = \left(\lim_{x\to a} f(x)\right)^{1/2} = 9^{1/2} = 3$ .

Now apply the product rule: $\lim_{x\to a} z(x) = \lim_{x\to a} [f(x)]^{1/2} \cdot \lim_{x\to a} g(x) = 3 \cdot 5 = 15$ .

Limit of a product involving a power rule

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