Sure, let's solve the limit [tex]$\lim _{x \rightarrow a} \frac{x^{2 / 3}-a^{2 / 3}}{x-a}$[/tex] step-by-step.
### Step-by-Step Solution:
#### Step 1: Identify the form of the limit
The limit we want to find is
[tex]$\lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a}.$[/tex]
Notice that if we try to directly substitute [tex]\( x = a \)[/tex], we get an indeterminate form of [tex]\( \frac{0}{0} \)[/tex].
#### Step 2: Use algebraic manipulation
To deal with the indeterminate form, we try to simplify the given expression. Here's a common technique:
Rewriting the numerator, consider the difference of powers formula in the generalized form:
[tex]$x^{k} - a^{k} = (x - a)(x^{k-1} + x^{k-2}a + \cdots + a^{k-1}).$[/tex]
In our problem, [tex]\( k = 2/3 \)[/tex],
[tex]$x^{2/3} - a^{2/3} = (x - a)\left(\frac{x^{2/3} - a^{2/3}}{x - a}\right).$[/tex]
This doesn't provide direct simplification. Thus, we look to other techniques such as L'Hopital's rule or a known series expansion.
#### Step 3: Applying standard forms or series (skip, directly apply L'Hopital if known form isn't derived)
Since direct techniques might still leave it complex, apply L'Hopital's Rule which is justified for [tex]\( \frac{0}{0} \)[/tex]:
We need the derivatives of the numerator and denominator. The numerator is [tex]\( x^{2/3} \)[/tex] and the derivative is:
[tex]$\frac{d}{dx}\left( x^{2/3} \right) = \frac{2}{3} x^{-1/3}.$[/tex]
The denominator [tex]\( x - a \)[/tex] has a derivative of:
[tex]$\frac{d}{dx}(x - a) = 1.$[/tex]
So, applying L'Hopital's Rule:
[tex]$ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a}
= \lim_{x \to a} \frac{\frac{2}{3} x^{-1/3}}{1} = \frac{2}{3} a^{-1/3} = \frac{2}{3 \cdot a^{1/3}}.$[/tex]
Thus, the solution to the limit is:
[tex]$ \boxed{\frac{2}{3a^{1/3}}}. $[/tex]
