To find the limit
[tex]\[ \lim _{x \rightarrow a} \frac{x^{\frac{7}{3}} - a^{\frac{7}{3}}}{x^{\frac{1}{2}} - a^{\frac{1}{2}}}, \][/tex]
we approach the problem as follows:
### Step 1: Recognize the Indeterminate Form
When [tex]\( x = a \)[/tex], both the numerator and the denominator become zero, giving us an indeterminate form of [tex]\(\frac{0}{0}\)[/tex]. Therefore, we can apply L'Hôpital's Rule, which states that:
[tex]\[
\lim_{{x \to a}} \frac{f(x)}{g(x)} = \lim_{{x \to a}} \frac{f'(x)}{g'(x)}
\][/tex]
provided the limit exists.
### Step 2: Identify [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
Here, [tex]\( f(x) = x^{\frac{7}{3}} - a^{\frac{7}{3}} \)[/tex] and [tex]\( g(x) = x^{\frac{1}{2}} - a^{\frac{1}{2}} \)[/tex].
### Step 3: Differentiate [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
First, find the derivative of [tex]\( f(x) \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}\left( x^{\frac{7}{3}} - a^{\frac{7}{3}} \right) = \frac{7}{3} x^{\frac{4}{3}}
\][/tex]
Second, find the derivative of [tex]\( g(x) \)[/tex]:
[tex]\[
g'(x) = \frac{d}{dx}\left( x^{\frac{1}{2}} - a^{\frac{1}{2}} \right) = \frac{1}{2} x^{-\frac{1}{2}}
\][/tex]
### Step 4: Apply L'Hôpital's Rule
According to L'Hôpital's Rule:
[tex]\[
\lim_{{x \to a}} \frac{x^{\frac{7}{3}} - a^{\frac{7}{3}}}{x^{\frac{1}{2}} - a^{\frac{1}{2}}} = \lim_{{x \to a}} \frac{\frac{7}{3} x^{\frac{4}{3}}}{\frac{1}{2} x^{-\frac{1}{2}}}
\][/tex]
### Step 5: Simplify the Expression
[tex]\[
\lim_{{x \to a}} \frac{\frac{7}{3} x^{\frac{4}{3}}}{\frac{1}{2} x^{-\frac{1}{2}}} = \lim_{{x \to a}} \frac{7 x^{\frac{4}{3} + \frac{1}{2}}}{3 \cdot \frac{1}{2}} = \lim_{{x \to a}} \frac{7 x^{\frac{4}{3} + \frac{1}{2}}}{\frac{3}{2}}
\][/tex]
Combine the exponents:
[tex]\[
\frac{4}{3} + \frac{1}{2} = \frac{8}{6} + \frac{3}{6} = \frac{11}{6}
\][/tex]
So the limit becomes:
[tex]\[
\lim_{{x \to a}} \frac{7 x^{\frac{11}{6}}}{\frac{3}{2}} = \lim_{{x \to a}} \frac{14 x^{\frac{11}{6}}}{3}
\][/tex]
### Step 6: Evaluate the Limit
Since [tex]\( x \to a \)[/tex]:
[tex]\[
\frac{14 x^{\frac{11}{6}}}{3} \longrightarrow \frac{14 a^{\frac{11}{6}}}{3}
\][/tex]
Thus, the limit is given by:
[tex]\[
\lim_{{x \to a}} \frac{x^{\frac{7}{3}} - a^{\frac{7}{3}}}{x^{\frac{1}{2}} - a^{\frac{1}{2}}} = \frac{14 a^{\frac{11}{6}}}{3}
\][/tex]
This is the final answer.
