Answer :
Of course! Let’s solve the limit problem step-by-step:
We want to find the limit:
[tex]\[ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a} \][/tex]
First, observe that directly substituting [tex]\( x = a \)[/tex] into the expression results in an indeterminate form of type [tex]\( \frac{0}{0} \)[/tex]. To resolve this, we can use the method of algebraic manipulation and properties of limits.
### Step 1: Rewrite the expression using the difference of powers
The expression [tex]\( x^{2/3} - a^{2/3} \)[/tex] can be rewritten in a form that will make it easier to work with. The idea is to use the fact that:
[tex]\[ x^{2/3} - a^{2/3} = (x^{1/3})^{2} - (a^{1/3})^{2} \][/tex]
Applying the difference of squares formula, [tex]\( A^2 - B^2 = (A - B)(A + B) \)[/tex], where [tex]\( A = x^{1/3} \)[/tex] and [tex]\( B = a^{1/3} \)[/tex]:
[tex]\[ (x^{1/3})^2 - (a^{1/3})^2 = (x^{1/3} - a^{1/3})(x^{1/3} + a^{1/3}) \][/tex]
### Step 2: Use this substitution in the original limit expression
Substitute the above expression into the limit:
[tex]\[ \frac{x^{2/3} - a^{2/3}}{x - a} = \frac{(x^{1/3} - a^{1/3})(x^{1/3} + a^{1/3})}{x - a} \][/tex]
### Step 3: Simplify the expression using a limit property
Now, observe the expression [tex]\(x^{1/3}\)[/tex] and [tex]\(a^{1/3}\)[/tex]. Note that:
[tex]\[ x^{1/3} - a^{1/3} \][/tex]
To further simplify, we can use a neat trick involving derivatives. Recall the definition of the derivative:
[tex]\[ \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a) \][/tex]
Here, let [tex]\( f(x) = x^{1/3} \)[/tex]. Then:
[tex]\[ f'(x) = \frac{d}{dx}\left( x^{1/3} \right) = \frac{1}{3} x^{-2/3} \][/tex]
Using the definition of the derivative at [tex]\( x = a \)[/tex] for [tex]\( f(x) = x^{1/3} \)[/tex]:
[tex]\[ \lim_{x \to a} \frac{x^{1/3} - a^{1/3}}{x - a} = \frac{1}{3} a^{-2/3} = \frac{1}{3a^{2/3}} \][/tex]
### Step 4: Apply the simplified result
Now, applying this in our simplified limit expression:
[tex]\[ \frac{(x^{1/3} - a^{1/3})(x^{1/3} + a^{1/3})}{x - a} \][/tex]
Given the above derivative limit property:
[tex]\[ \lim_{x \to a} \frac{x^{1/3} - a^{1/3}}{x - a} = \frac{1}{3a^{2/3}} \][/tex]
Thus:
[tex]\[ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a} = \left( \frac{1}{3a^{2/3}} \right) \cdot (a^{1/3} + a^{1/3}) = \left( \frac{1}{3a^{2/3}} \right) \cdot (2a^{1/3}) \][/tex]
### Conclusion
Finally, we simplify this result:
[tex]\[ \left( \frac{1}{3a^{2/3}} \right) \cdot (2a^{1/3}) = \frac{2a^{1/3}}{3a^{2/3}} = \frac{2a^{1/3}}{3a^{2/3}} = \frac{2}{3} \cdot a^{-1/3} = \frac{2}{3a^{1/3}} \][/tex]
Therefore, the limit is:
[tex]\[ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a} = \frac{2}{3a^{1/3}} \][/tex]
We want to find the limit:
[tex]\[ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a} \][/tex]
First, observe that directly substituting [tex]\( x = a \)[/tex] into the expression results in an indeterminate form of type [tex]\( \frac{0}{0} \)[/tex]. To resolve this, we can use the method of algebraic manipulation and properties of limits.
### Step 1: Rewrite the expression using the difference of powers
The expression [tex]\( x^{2/3} - a^{2/3} \)[/tex] can be rewritten in a form that will make it easier to work with. The idea is to use the fact that:
[tex]\[ x^{2/3} - a^{2/3} = (x^{1/3})^{2} - (a^{1/3})^{2} \][/tex]
Applying the difference of squares formula, [tex]\( A^2 - B^2 = (A - B)(A + B) \)[/tex], where [tex]\( A = x^{1/3} \)[/tex] and [tex]\( B = a^{1/3} \)[/tex]:
[tex]\[ (x^{1/3})^2 - (a^{1/3})^2 = (x^{1/3} - a^{1/3})(x^{1/3} + a^{1/3}) \][/tex]
### Step 2: Use this substitution in the original limit expression
Substitute the above expression into the limit:
[tex]\[ \frac{x^{2/3} - a^{2/3}}{x - a} = \frac{(x^{1/3} - a^{1/3})(x^{1/3} + a^{1/3})}{x - a} \][/tex]
### Step 3: Simplify the expression using a limit property
Now, observe the expression [tex]\(x^{1/3}\)[/tex] and [tex]\(a^{1/3}\)[/tex]. Note that:
[tex]\[ x^{1/3} - a^{1/3} \][/tex]
To further simplify, we can use a neat trick involving derivatives. Recall the definition of the derivative:
[tex]\[ \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a) \][/tex]
Here, let [tex]\( f(x) = x^{1/3} \)[/tex]. Then:
[tex]\[ f'(x) = \frac{d}{dx}\left( x^{1/3} \right) = \frac{1}{3} x^{-2/3} \][/tex]
Using the definition of the derivative at [tex]\( x = a \)[/tex] for [tex]\( f(x) = x^{1/3} \)[/tex]:
[tex]\[ \lim_{x \to a} \frac{x^{1/3} - a^{1/3}}{x - a} = \frac{1}{3} a^{-2/3} = \frac{1}{3a^{2/3}} \][/tex]
### Step 4: Apply the simplified result
Now, applying this in our simplified limit expression:
[tex]\[ \frac{(x^{1/3} - a^{1/3})(x^{1/3} + a^{1/3})}{x - a} \][/tex]
Given the above derivative limit property:
[tex]\[ \lim_{x \to a} \frac{x^{1/3} - a^{1/3}}{x - a} = \frac{1}{3a^{2/3}} \][/tex]
Thus:
[tex]\[ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a} = \left( \frac{1}{3a^{2/3}} \right) \cdot (a^{1/3} + a^{1/3}) = \left( \frac{1}{3a^{2/3}} \right) \cdot (2a^{1/3}) \][/tex]
### Conclusion
Finally, we simplify this result:
[tex]\[ \left( \frac{1}{3a^{2/3}} \right) \cdot (2a^{1/3}) = \frac{2a^{1/3}}{3a^{2/3}} = \frac{2a^{1/3}}{3a^{2/3}} = \frac{2}{3} \cdot a^{-1/3} = \frac{2}{3a^{1/3}} \][/tex]
Therefore, the limit is:
[tex]\[ \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x - a} = \frac{2}{3a^{1/3}} \][/tex]
