Question

question 16 (2 points)

determine, using the properties of limits:

\\\lim_{x \to 0} \frac{(x+64)^{\frac{1}{3}} - 4}{x}\\

\\(\bigcirc\\) \\(\frac{1}{48}\\)

\\(\bigcirc\\) \\(\frac{1}{16}\\)

\\(\bigcirc\\) \\(\frac{1}{64}\\)

\\(\bigcirc\\) 4

Explanation:

Identify the limit type

Using the Limit Evaluation Techniques knowledge point

$$ \lim_{x \to 0} \frac{(x+64)^{\frac{1}{3}} - 4}{x} $$

Direct substitution yields the indeterminate form \(\frac{0}{0}\).

Apply algebraic substitution

Using the Limit Evaluation Techniques knowledge point
Let \(u = (x+64)^{\frac{1}{3}}\).
As \(x \to 0\), \(u \to 64^{\frac{1}{3}} = 4\).
Since \(u^3 = x + 64\), we have \(x = u^3 - 64\).

Rewrite the limit expression

Using the Properties of Limits knowledge point

$$ \lim_{u \to 4} \frac{u - 4}{u^3 - 64} $$

Factor the denominator

Using the Limit Evaluation Techniques knowledge point

$$ u^3 - 64 = (u - 4)(u^2 + 4u + 16) $$

Thus, the limit becomes:

$$ \lim_{u \to 4} \frac{u - 4}{(u - 4)(u^2 + 4u + 16)} = \lim_{u \to 4} \frac{1}{u^2 + 4u + 16} $$

Evaluate the simplified limit

Using the Properties of Limits knowledge point

$$ \frac{1}{4^2 + 4(4) + 16} = \frac{1}{16 + 16 + 16} = \frac{1}{48} $$

Answer:

• (A) \(\frac{1}{48}\) (Correct answer)
• (B) \(\frac{1}{16}\)
• (C) \(\frac{1}{64}\)
• (D) 4