To evaluate the limit [tex]\(\lim_{x \to a} \frac{x^{\frac{2}{3}} - a^{\frac{2}{3}}}{x - a}\)[/tex], we can utilize the concept of limits and L'Hôpital's Rule, or alternatively, we can simplify using algebraic manipulation. Here's a step-by-step solution:
1. Identify the Indeterminate Form:
First, observe that if you directly substitute [tex]\( x = a \)[/tex] into the expression [tex]\(\frac{x^{\frac{2}{3}} - a^{\frac{2}{3}}}{x - a}\)[/tex], you get:
[tex]\[
\frac{a^{\frac{2}{3}} - a^{\frac{2}{3}}}{a - a} = \frac{0}{0}
\][/tex]
This is an indeterminate form, which means we need to perform more sophisticated analysis to evaluate the limit.
2. Apply L'Hôpital's Rule:
L'Hôpital's Rule states that for limits of the form [tex]\(\frac{0}{0}\)[/tex] or [tex]\(\frac{\infty}{\infty}\)[/tex]:
[tex]\[
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
\][/tex]
provided the limit on the right side exists. Here, let [tex]\(f(x) = x^{\frac{2}{3}} - a^{\frac{2}{3}}\)[/tex] and [tex]\(g(x) = x - a\)[/tex].
3. Differentiate the Numerator and Denominator:
- Differentiate the numerator [tex]\( f(x) = x^{\frac{2}{3}} - a^{\frac{2}{3}} \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx} \left( x^{\frac{2}{3}} \right) = \frac{2}{3} x^{-\frac{1}{3}}
\][/tex]
Since [tex]\(a^{\frac{2}{3}}\)[/tex] is a constant and its derivative is 0.
- Differentiate the denominator [tex]\( g(x) = x - a \)[/tex]:
[tex]\[
g'(x) = \frac{d}{dx} \left( x - a \right) = 1
\][/tex]
4. Evaluate the Limit of the Derivatives:
Apply the derivatives back into the limit:
[tex]\[
\lim_{x \to a} \frac{f'(x)}{g'(x)} = \lim_{x \to a} \frac{\frac{2}{3} x^{-\frac{1}{3}}}{1} = \lim_{x \to a} \frac{2}{3} x^{-\frac{1}{3}}
\][/tex]
Now, substitute [tex]\( x = a \)[/tex]:
[tex]\[
\frac{2}{3} a^{-\frac{1}{3}}
\][/tex]
5. Simplify the Result:
The expression [tex]\(a^{-\frac{1}{3}}\)[/tex] can be rewritten as [tex]\(\frac{1}{a^{\frac{1}{3}}}\)[/tex]. Therefore:
[tex]\[
\frac{2}{3} a^{-\frac{1}{3}} = \frac{2}{3} \cdot \frac{1}{a^{\frac{1}{3}}} = \frac{2}{3 a^{\frac{1}{3}}}
\][/tex]
Hence, the evaluated limit is:
[tex]\[
\lim_{x \to a} \frac{x^{\frac{2}{3}} - a^{\frac{2}{3}}}{x - a} = \frac{2}{3 a^{\frac{1}{3}}}
\][/tex]
