To find the limit [tex]\( \lim_{{x \to a}} \frac{x^{1/3} - a^{1/3}}{x^{1/2} - a^{1/2}} \)[/tex], we'll go through a detailed, step-by-step solution.
First, we observe that the expression is indeterminate of the form [tex]\( \frac{0}{0} \)[/tex] when [tex]\( x = a \)[/tex]. Therefore, we can apply L'Hôpital's Rule, which is useful for handling such indeterminate forms.
1. Identify the numerator and the denominator:
- Numerator: [tex]\( x^{1/3} - a^{1/3} \)[/tex]
- Denominator: [tex]\( x^{1/2} - a^{1/2} \)[/tex]
2. Differentiate the numerator and the denominator separately:
- The derivative of the numerator [tex]\( x^{1/3} - a^{1/3} \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d}{dx} (x^{1/3}) = \frac{1}{3} x^{-2/3} = \frac{1}{3} x^{-\frac{2}{3}}
\][/tex]
(Note: Since [tex]\( a^{1/3} \)[/tex] is a constant, its derivative is 0.)
- The derivative of the denominator [tex]\( x^{1/2} - a^{1/2} \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d}{dx} (x^{1/2}) = \frac{1}{2} x^{-1/2} = \frac{1}{2} x^{-\frac{1}{2}}
\][/tex]
(Again, since [tex]\( a^{1/2} \)[/tex] is a constant, its derivative is 0.)
3. Apply L'Hôpital's Rule:
According to L'Hôpital's Rule, for the limit of the form [tex]\( \frac{0}{0} \)[/tex], we have:
[tex]\[
\lim_{{x \to a}} \frac{x^{1/3} - a^{1/3}}{x^{1/2} - a^{1/2}} = \lim_{{x \to a}} \frac{\frac{d}{dx}(x^{1/3})}{\frac{d}{dx}(x^{1/2})} = \lim_{{x \to a}} \frac{\frac{1}{3} x^{-\frac{2}{3}}}{\frac{1}{2} x^{-\frac{1}{2}}}
\][/tex]
4. Simplify the resulting expression:
[tex]\[
\lim_{{x \to a}} \frac{\frac{1}{3} x^{-\frac{2}{3}}}{\frac{1}{2} x^{-\frac{1}{2}}} = \lim_{{x \to a}} \frac{\frac{1}{3} x^{-\frac{2}{3}} \cdot 2 x^{\frac{1/2}}}{1} = \lim_{{x \to a}} \frac{2}{3} x^{\frac{-2}{3} + \frac{1}{2}}
\][/tex]
Simplify the exponent:
[tex]\[
\frac{2}{3} x^{{-\frac{2}{3} + \frac{1}{2}}} = \frac{2}{3} x^{\frac{-4}{6} + \frac{3}{6}} = \frac{2}{3} x^{-\frac{1}{6}}
\][/tex]
5. Substitute [tex]\( x = a \)[/tex] into the simplified expression:
[tex]\[
\frac{2}{3} a^{-\frac{1}{6}}
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{\frac{2}{3} a^{-\frac{1}{6}}}
\][/tex]
