To evaluate the limit [tex]\(\lim_{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4}\)[/tex], let's proceed step-by-step.
### Step 1: Factor the numerator and the denominator
First, we'll use difference of powers factorizations for both the numerator and the denominator.
1. Numerator: The expression [tex]\(x^5 - a^5\)[/tex] can be factored using the difference of powers formula:
[tex]\[
x^5 - a^5 = (x - a)(x^4 + x^3 a + x^2 a^2 + x a^3 + a^4)
\][/tex]
2. Denominator: The expression [tex]\(x^4 - a^4\)[/tex] can also be factored using the difference of squares formula iteratively, but it's simpler to use the difference of powers directly:
[tex]\[
x^4 - a^4 = (x - a)(x^3 + x^2 a + x a^2 + a^3)
\][/tex]
### Step 2: Simplify the expression
After factoring, we substitute these factorizations into the limit:
[tex]\[
\lim_{x \rightarrow a} \frac{(x - a)(x^4 + x^3 a + x^2 a^2 + x a^3 + a^4)}{(x - a)(x^3 + x^2 a + x a^2 + a^3)}
\][/tex]
We notice that the terms [tex]\((x - a)\)[/tex] cancel out:
[tex]\[
\lim_{x \rightarrow a} \frac{x^4 + x^3 a + x^2 a^2 + x a^3 + a^4}{x^3 + x^2 a + x a^2 + a^3}
\][/tex]
### Step 3: Evaluate the limit
Now that the expression is simplified, we can directly substitute [tex]\(x = a\)[/tex]:
[tex]\[
\frac{a^4 + a^3 a + a^2 a^2 + a a^3 + a^4}{a^3 + a^2 a + a a^2 + a^3}
\][/tex]
Simplifying the numerator and the denominator:
- Numerator:
[tex]\[
a^4 + a^4 + a^4 + a^4 + a^4 = 5a^4
\][/tex]
- Denominator:
[tex]\[
a^3 + a^3 + a^3 + a^3 = 4a^3
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\frac{5a^4}{4a^3} = \frac{5}{4} a
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{\frac{5}{4} a}
\][/tex]
