To solve the limit [tex]\(\lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4}\)[/tex], we need to consider the behavior of the given expression as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]. Let's go through the solution step-by-step.
First, rewrite the given limit expression:
[tex]\[
\lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4}
\][/tex]
Observe that both the numerator and the denominator become indeterminate forms [tex]\(0/0\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]. To simplify this expression, we can factorize the numerator and denominator.
### Step 1: Factorize the Numerator and Denominator
The expression in the numerator [tex]\(x^5 - a^5\)[/tex] can be factored using the difference of powers:
[tex]\[
x^5 - a^5 = (x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4)
\][/tex]
Similarly, the expression in the denominator [tex]\(x^4 - a^4\)[/tex] can be factored using the difference of squares twice:
[tex]\[
x^4 - a^4 = (x^2 - a^2)(x^2 + a^2) = (x - a)(x + a)(x^2 + a^2)
\][/tex]
Thus, we can rewrite the limit expression as:
[tex]\[
\frac{x^5 - a^5}{x^4 - a^4} = \frac{(x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4)}{(x - a)(x + a)(x^2 + a^2)}
\][/tex]
### Step 2: Cancel the Common Factor
Since [tex]\(x \neq a\)[/tex] (as we are just looking at the limit approaching [tex]\(a\)[/tex]), we can cancel out the common factor [tex]\((x - a)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{x^4 + x^3a + x^2a^2 + xa^3 + a^4}{(x + a)(x^2 + a^2)}
\][/tex]
### Step 3: Evaluate the Limit as [tex]\(x\)[/tex] Approaches [tex]\(a\)[/tex]
Now, substitute [tex]\(x = a\)[/tex] in the simplified expression to evaluate the limit:
[tex]\[
\lim _{x \rightarrow a} \frac{x^4 + x^3a + x^2a^2 + xa^3 + a^4}{(x + a)(x^2 + a^2)}
\][/tex]
Substituting [tex]\(x = a\)[/tex]:
[tex]\[
= \frac{a^4 + a^4 + a^4 + a^4 + a^4}{(a + a)(a^2 + a^2)}
\][/tex]
[tex]\[
= \frac{5a^4}{2a \cdot 2a^2}
\][/tex]
[tex]\[
= \frac{5a^4}{4a^3}
\][/tex]
[tex]\[
= \frac{5a^4}{4a^3} = \frac{5a}{4}
\][/tex]
So, the limit is:
[tex]\[
\lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4} = \frac{5a}{4}
\][/tex]
Thus, the solution to the limit is:
[tex]\[
\boxed{\frac{5a}{4}}
\][/tex]
