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Evaluate the limit:

[tex]\[ \lim _{x \rightarrow 64} \frac{x^{1 / 6}-2}{x^{1 / 3}-4} \][/tex]


Sagot :

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]