Question

assume \\( \lim_{x\to 5} f(x) = 6 \\) and \\( \lim_{x\to 5} g(x) = 9 \\). compute the following limit and state the limit laws used to justify the computation. \\( \lim_{x\to 5} \sqrt3{f(x)g(x) + 10} = \\) (simplify your answer.) select each limit law used to justify the computation. a. difference b. root c. product d. constant multiple e. sum f. power g. quotient

Explanation:

Step1: Apply Product Law

First, we use the product law of limits, which states that \(\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\) (when both limits exist). So, \(\lim_{x \to 5} [f(x)g(x)]=\lim_{x \to 5} f(x) \cdot \lim_{x \to 5} g(x)=6\times9 = 54\).

Step2: Apply Sum Law

Next, use the sum law of limits: \(\lim_{x \to a} [h(x)+k(x)]=\lim_{x \to a} h(x)+\lim_{x \to a} k(x)\) (when both limits exist). Here, \(h(x)=f(x)g(x)\) and \(k(x) = 10\) (a constant, so \(\lim_{x \to 5}10 = 10\)). Thus, \(\lim_{x \to 5}[f(x)g(x)+10]=\lim_{x \to 5}[f(x)g(x)]+\lim_{x \to 5}10=54 + 10=64\).

Step3: Apply Root Law

Finally, use the root law of limits: \(\lim_{x \to a}\sqrt[n]{h(x)}=\sqrt[n]{\lim_{x \to a}h(x)}\) (when \(\lim_{x \to a}h(x)\) exists and for even \(n\), \(\lim_{x \to a}h(x)\geq0\)). For \(n = 3\) (odd), we have \(\lim_{x \to 5}\sqrt[3]{f(x)g(x)+10}=\sqrt[3]{\lim_{x \to 5}[f(x)g(x)+10]}=\sqrt[3]{64}\).

Step4: Simplify the Cube Root

Simplify \(\sqrt[3]{64}\). Since \(4^3=64\), \(\sqrt[3]{64} = 4\).

Answer:

The value of the limit \(\lim_{x \to 5}\sqrt[3]{f(x)g(x)+10}\) is \(4\). The limit laws used are Product (C), Sum (E), and Root (B).