To evaluate the limit [tex]\(\lim _{n \rightarrow b} \frac{n^2 - b^2}{n^{4 / 3} - b^{4 / 3}}\)[/tex], let's proceed step-by-step:
1. Recognize the form:
The limit is of the form where both the numerator [tex]\(n^2 - b^2\)[/tex] and the denominator [tex]\(n^{4/3} - b^{4/3}\)[/tex] become zero as [tex]\(n\)[/tex] approaches [tex]\(b\)[/tex]. This is a [tex]\( \frac{0}{0} \)[/tex] indeterminate form, and we will need to apply some algebraic techniques or calculus techniques like L'Hôpital's rule to resolve this.
2. Factor the numerator and denominator:
The numerator [tex]\(n^2 - b^2\)[/tex] can be factored using the difference of squares:
[tex]\[
n^2 - b^2 = (n - b)(n + b)
\][/tex]
3. Write the expression with the factors:
Substituting the factored form of the numerator into the limit expression, we get:
[tex]\[
\lim _{n \rightarrow b} \frac{(n - b)(n + b)}{n^{4 / 3} - b^{4 / 3}}
\][/tex]
4. Simplifying the expression:
For the denominator [tex]\(n^{4/3} - b^{4/3}\)[/tex], it doesn't factor neatly, so we will look to simplify using some other method.
5. Using calculus techniques:
Instead of trying to factor the denominator directly, we differentiate the numerator and denominator:
- The derivative of the numerator [tex]\(n^2 - b^2\)[/tex] with respect to [tex]\(n\)[/tex] is:
[tex]\[
\frac{d}{dn} (n^2 - b^2) = 2n
\][/tex]
- The derivative of the denominator [tex]\(n^{4/3} - b^{4/3}\)[/tex] with respect to [tex]\(n\)[/tex] is:
[tex]\[
\frac{d}{dn} (n^{4/3} - b^{4/3}) = \frac{4}{3} n^{1/3}
\][/tex]
6. Applying L'Hôpital’s Rule:
Since both the numerator and the denominator give 0 as [tex]\(n \to b\)[/tex], we apply L'Hôpital’s rule:
[tex]\[
\lim _{n \rightarrow b} \frac{(n - b)(n + b)}{n^{4 / 3} - b^{4 / 3}} = \lim _{n \rightarrow b} \frac{2n}{\frac{4}{3} n^{1/3}}
\][/tex]
7. Simplifying the limit:
The simplified form is:
[tex]\[
\lim _{n \rightarrow b} \frac{2n}{\frac{4}{3} n^{1/3}} = \lim _{n \rightarrow b} \frac{2n \cdot 3}{4 n^{1/3}} = \lim _{n \rightarrow b} \frac{6n}{4 n^{1/3}} = \frac{3}{2} \lim _{n \rightarrow b} \frac{n^{2/3}}{1}
\][/tex]
8. Evaluating the limit:
Finally, as [tex]\(n \to b\)[/tex], the expression [tex]\( \frac{n^{2/3}}{1} \)[/tex] simply becomes [tex]\( b^{2/3} \)[/tex]:
[tex]\[
\frac{3}{2} \cdot b^{2/3}
\][/tex]
Thus, the limit is:
[tex]\[
\lim _{n \rightarrow b} \frac{n^2 - b^2}{n^{4 / 3} - b^{4 / 3}} = \frac{3}{2} b^{2/3}
\][/tex]
