Question

classify the statement as either true or false
if \\( \lim_{x \to 1} g(x) = 3 \\), then \\( \lim_{x \to 1} g(x)^2 = 9 \\)

choose the correct answer below
\\( \bigcirc \\) a. the statement is false because the given value of the limit \\( \lim_{x \to 1} g(x)^2 \\) is incorrect according to the limit properties
\\( \bigcirc \\) b. the statement is true because the given value of the limit \\( \lim_{x \to 1} g(x)^2 \\) is correct according to the limit properties
\\( \bigcirc \\) c. the statement is true because the given value of the limit \\( \lim_{x \to 1} g(x)^2 \\) is incorrect according to the limit properties
\\( \bigcirc \\) d. the statement is false because the given value of the limit \\( \lim_{x \to 1} g(x)^2 \\) is correct according to the limit properties

Explanation:

Step1: Recall limit power rule

If $\lim_{x \to a} f(x) = L$, then $\lim_{x \to a} [f(x)]^n = L^n$.

Step2: Apply rule to given limit

$\lim_{x \to 1} g(x) = 3$, so $\lim_{x \to 1} [g(x)]^2 = 3^2 = 9$.

Step3: Evaluate the statement

The value matches the rule, so the statement is true.

Answer:

B. The statement is true because the given value of the limit $\lim_{x \to 1} [g(x)]^2$ is correct according to the limit properties.