We are tasked with evaluating the following limit:

$\lim_{x \to 64} \left( \frac{\sqrt{x^6} - 2}{\sqrt{x^3} - 4} \right)$

Step 1: Substitute $x = 64$ directly into the expression

We first check the direct substitution:

• $\sqrt{x^6} = (x^6)^{1/2} = x^3$, so $\sqrt{64^6} = 64^3 = 262144$.
• $\sqrt{x^3} = (x^3)^{1/2} = x^{3/2}$, so $\sqrt{64^3} = 64^{3/2} = 512$.

Substituting $x = 64$ into the expression:

$\frac{\sqrt{64^6} - 2}{\sqrt{64^3} - 4} = \frac{262144 - 2}{512 - 4} = \frac{262142}{508}$

Thus, the value of the limit is:

$\frac{262142}{508} = 515$

Final Answer:

$\lim_{x \to 64} \left( \frac{\sqrt{x^6} - 2}{\sqrt{x^3} - 4} \right) = 515$

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If you'd like a more detailed breakdown of the steps or further explanations, feel free to ask! Here are a few related questions to expand your understanding:

1. How can we simplify similar expressions involving square roots and powers?
2. What other techniques can be used to evaluate limits involving indeterminate forms?
3. How does the limit change if the denominator is a different function of $x$?
4. What role does continuity play in evaluating limits?
5. How can we apply the Squeeze Theorem to limit problems?

Tip: Whenever you get an indeterminate form like $0/0$ when evaluating limits, consider using algebraic manipulation, such as factoring or rationalizing the expression, to simplify it.