To find the limit of the expression [tex]\(\lim_{x \rightarrow 64} \frac{x^{1/6} - 2}{x^{1/3} - 4}\)[/tex], let's go through the steps:
1. Recognize the form: First, substitute [tex]\( x = 64 \)[/tex] into the expression to observe the form.
[tex]\[
\frac{64^{1/6} - 2}{64^{1/3} - 4}
\][/tex]
Calculate [tex]\( 64^{1/6} \)[/tex] and [tex]\( 64^{1/3} \)[/tex]:
- [tex]\( 64^{1/6} = 2 \)[/tex] (since [tex]\( 64 = 2^6 \)[/tex])
- [tex]\( 64^{1/3} = 4 \)[/tex] (since [tex]\( 64 = 4^3 \)[/tex])
This gives us:
[tex]\[
\frac{2 - 2}{4 - 4} = \frac{0}{0}
\][/tex]
which is an indeterminate form. Thus, we need to apply another method to evaluate the limit.
2. Rewrite using substitution: Let [tex]\( u = x^{1/3} \)[/tex]. Therefore, [tex]\( u^3 = x \)[/tex] and as [tex]\( x \to 64 \)[/tex], [tex]\( u \to 4 \)[/tex].
The expression now becomes:
[tex]\[
\frac{(u^3)^{1/6} - 2}{u - 4} = \frac{u^{1/2} - 2}{u - 4}
\][/tex]
3. Identify another substitution: Let [tex]\( v = u - 4 \)[/tex], so when [tex]\( u \to 4 \)[/tex], [tex]\( v \to 0 \)[/tex]. The expression transforms as follows:
[tex]\[
u = v + 4
\][/tex]
Hence,
[tex]\[
\frac{(4 + v)^{1/2} - 2}{v}
\][/tex]
4. Expand using a binomial approximation: For small [tex]\( v \)[/tex], [tex]\((4 + v)^{1/2}\)[/tex] can be approximated using the binomial expansion around [tex]\( v = 0 \)[/tex]:
[tex]\[
(4+v)^{1/2} \approx 2 + \frac{v}{4}
\][/tex]
Therefore:
[tex]\[
\frac{(2 + \frac{v}{4}) - 2}{v} = \frac{\frac{v}{4}}{v} = \frac{1}{4}
\][/tex]
5. Calculate the limit: As [tex]\( v \to 0 \)[/tex], the expression simplifies to:
[tex]\[
\lim_{v \to 0} \frac{1}{4} = \frac{1}{4}
\][/tex]
Thus, the limit is:
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
