Question

compute the following limit assuming \\(\lim_{x \to 2} f(x) = 8\\). state the limit law(s) used to justify the computation.

\\(\lim_{x \to 2} 2f(x)\\)

choose the correct answer below.

a. \\(\lim_{x \to 2} 2f(x) = \\) by power law of limit.
(simplify your answer.)

b. \\(\lim_{x \to 2} 2f(x) = \\) by constant multiple law of limit.
(simplify your answer.)

Explanation:

Identify given values and target

Given:

$$ \lim_{x \to 2} f(x) = 8 $$

Target:

$$ \lim_{x \to 2} [2f(x)] $$

Apply the constant multiple law of limits

$$ \lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x) $$

Substitute \(k = 2\) and \(c = 2\):

$$ \lim_{x \to 2} [2f(x)] = 2 \cdot \lim_{x \to 2} f(x) $$

Calculate the final value

$$ 2 \cdot 8 = 16 $$

This matches option B.

Answer:

• A. \(\lim_{x \to 2} [2f(x)] = 16\) by power law of limit.
• B. \(\lim_{x \to 2} [2f(x)] = 16\) by constant multiple law of limit. (Correct answer)