To solve for 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], let's follow a step-by-step approach using calculus techniques:
1. Expression Analysis:
We need to determine the behavior of the given expression as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex].
[tex]\[
\lim_{x \rightarrow a} \frac{x^{\frac{7}{3}} - a^{\frac{7}{3}}}{x^{\frac{1}{2}} - a^{\frac{1}{2}}}
\][/tex]
2. Indeterminate Form:
Direct substitution [tex]\(x = a\)[/tex] in the expression results in the indeterminate form [tex]\( \frac{0}{0} \)[/tex], so we need to simplify or manipulate it to resolve the indeterminate form.
3. Use of L'Hospital's Rule:
Since the expression is in the indeterminate form [tex]\(\frac{0}{0}\)[/tex], L'Hospital's Rule is applicable. L'Hospital's Rule states that:
[tex]\[
\lim_{x \rightarrow a} \frac{f(x)}{g(x)} \text{ can be evaluated as } \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}
\][/tex]
provided the limit exists. In our case,
[tex]\[
f(x) = x^{\frac{7}{3}} - a^{\frac{7}{3}} \quad \text{and} \quad g(x) = x^{\frac{1}{2}} - a^{\frac{1}{2}}
\][/tex]
4. Differentiate the Numerator and Denominator:
Find [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]
Find [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]
5. Applying L'Hospital's Rule:
Applying L'Hospital's Rule to our limit:
[tex]\[
\lim_{x \rightarrow a} \frac{x^{\frac{7}{3}} - a^{\frac{7}{3}}}{x^{\frac{1}{2}} - a^{\frac{1}{2}}} = \lim_{x \rightarrow a} \frac{\frac{7}{3} x^{\frac{4}{3}}}{\frac{1}{2} x^{-\frac{1}{2}}}
\][/tex]
6. Simplify the Expression:
Simplify the new expression obtained after differentiation:
[tex]\[
\frac{\frac{7}{3} x^{\frac{4}{3}}}{\frac{1}{2} x^{-\frac{1}{2}}} = \frac{\frac{7}{3} x^{\frac{4}{3}} \cdot 2}{x^{-\frac{1}{2}}} = \frac{7 \cdot 2}{3} \cdot x^{\frac{4}{3} + \frac{1}{2}} = \frac{14}{3} x^{\frac{4}{3} + \frac{1}{2}}
\][/tex]
7. Combine Exponents:
Combine the exponents:
[tex]\[
x^{\frac{4}{3} + \frac{1}{2}} = x^{\frac{4}{3} + \frac{3}{6}} = x^{\frac{4}{3} + \frac{1}{2}} = x^{\frac{4}{3} + \frac{1}{2}} = x^{\frac{8}{6} + \frac{3}{6}} = x^{\frac{11}{6}}
\][/tex]
8. Final Limit Calculation:
Substitute [tex]\(x = a\)[/tex] into the simplified expression:
[tex]\[
\lim_{x \rightarrow a} \frac{14}{3} x^{\frac{11}{6}} = \frac{14}{3} a^{\frac{11}{6}}
\][/tex]
So, the limit is:
[tex]\[
\boxed{\frac{14}{3} a^{\frac{11}{6}}}
\][/tex]
