To evaluate the limit [tex]\(\lim_{{x \rightarrow a}} \frac{x^{2 / 3} - a^{2 / 3}}{x - a}\)[/tex], we'll use a combination of substitution and the properties of derivatives. Here is the detailed step-by-step solution:
1. Identify the Form of the Limit:
The given limit is of the form [tex]\(\frac{f(x) - f(a)}{x - a}\)[/tex], where [tex]\(f(x) = x^{2/3}\)[/tex]. This resembles the definition of the derivative of a function at a point.
2. Rewriting the Limit as a Derivative:
Recall that the derivative of a function [tex]\(f(x)\)[/tex] at a point [tex]\(a\)[/tex] is given by:
[tex]\[
f'(a) = \lim_{{x \rightarrow a}} \frac{f(x) - f(a)}{x - a}
\][/tex]
This matches the form of our limit expression. Therefore, evaluating the limit is equivalent to finding the derivative of [tex]\(f(x) = x^{2/3}\)[/tex] at [tex]\(x = a\)[/tex].
3. Derivative of [tex]\(f(x) = x^{2/3}\)[/tex]:
We need to find [tex]\(f'(x)\)[/tex], the derivative of [tex]\(x^{2/3}\)[/tex].
Using the power rule for derivatives, which states that [tex]\( \frac{d}{dx} x^n = n x^{n-1} \)[/tex], we get:
[tex]\[
f'(x) = \frac{d}{dx} x^{2/3} = \frac{2}{3} x^{(2/3)-1} = \frac{2}{3} x^{-1/3}
\][/tex]
4. Evaluate the Derivative at [tex]\(x = a\)[/tex]:
Substitute [tex]\(x = a\)[/tex] into the derivative:
[tex]\[
f'(a) = \frac{2}{3} a^{-1/3}
\][/tex]
5. Simplify:
[tex]\[
f'(a) = \frac{2}{3} \cdot \frac{1}{a^{1/3}} = \frac{2}{3a^{1/3}}
\][/tex]
Thus, the limit [tex]\(\lim_{{x \rightarrow a}} \frac{x^{2/3} - a^{2/3}}{x - a}\)[/tex] is [tex]\(\frac{2}{3a^{1/3}}\)[/tex].
