To find the limit [tex]\(\lim _{x \rightarrow 64} \frac{\sqrt[6]{x}-2}{\sqrt[3]{x}-4}\)[/tex], we proceed with the following steps:
1. Substitute [tex]\(x = 64\)[/tex] into the original limit expression:
[tex]\[
\frac{\sqrt[6]{64} - 2}{\sqrt[3]{64} - 4}
\][/tex]
Calculating each part separately at [tex]\(x = 64\)[/tex]:
- [tex]\(\sqrt[6]{64}\)[/tex] is [tex]\(\sqrt[6]{2^6} = 2\)[/tex].
- [tex]\(\sqrt[3]{64}\)[/tex] is [tex]\(\sqrt[3]{2^6} = 4\)[/tex].
Substituting these values in, we get:
[tex]\[
\frac{2 - 2}{4 - 4} = \frac{0}{0}
\][/tex]
This is an indeterminate form, indicating that further work is needed to simplify the expression.
2. Rewrite the limit expression and analyze the behavior as [tex]\(x\)[/tex] approaches 64:
The given expression is:
[tex]\[
\frac{\sqrt[6]{x} - 2}{\sqrt[3]{x} - 4}
\][/tex]
Let's denote [tex]\(y = \sqrt[3]{x}\)[/tex]. Then, [tex]\(x = y^3\)[/tex], and as [tex]\(x \to 64\)[/tex], [tex]\(y \to 4\)[/tex].
3. Rewrite the expression in terms of [tex]\(y\)[/tex]:
Substitute [tex]\(x = y^3\)[/tex]:
[tex]\[
\frac{\sqrt[6]{y^3} - 2}{y - 4}
\][/tex]
Since [tex]\(\sqrt[6]{y^3} = y^{1/2}\)[/tex], the expression becomes:
[tex]\[
\frac{y^{1/2} - 2}{y - 4}
\][/tex]
4. Simplify the transformed expression:
Consider the limit:
[tex]\[
\lim_{y \to 4} \frac{y^{1/2} - 2}{y - 4}
\][/tex]
To solve this limit, recognize that it is still in the [tex]\(\frac{0}{0}\)[/tex] indeterminate form, so we can use L'Hôpital's Rule, which requires differentiating the numerator and the denominator.
5. Apply L'Hôpital's Rule:
Differentiate the numerator and the denominator with respect to [tex]\(y\)[/tex]:
- Derivative of the numerator, [tex]\(y^{1/2} - 2\)[/tex], is [tex]\(\frac{1}{2} y^{-1/2}\)[/tex].
- Derivative of the denominator, [tex]\(y - 4\)[/tex], is [tex]\(1\)[/tex].
Thus, the limit becomes:
[tex]\[
\lim_{y \to 4} \frac{\frac{1}{2} y^{-1/2}}{1} = \lim_{y \to 4} \frac{1}{2} y^{-1/2}
\][/tex]
6. Evaluate the simplified limit:
Substitute [tex]\(y = 4\)[/tex]:
[tex]\[
\frac{1}{2} (4)^{-1/2} = \frac{1}{2} \cdot \frac{1}{\sqrt{4}} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
