To solve the limit
[tex]\[
\lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4},
\][/tex]
we can proceed by simplifying the expression step-by-step.
### Step 1: Factor the numerator and the denominator
Both the numerator, [tex]\(x^5 - a^5\)[/tex], and the denominator, [tex]\(x^4 - a^4\)[/tex], can be factored using known algebraic identities.
#### Factor the numerator [tex]\(x^5 - a^5\)[/tex]:
The numerator can be factored using the difference of powers formula:
[tex]\[
x^5 - a^5 = (x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4).
\][/tex]
#### Factor the denominator [tex]\(x^4 - a^4\)[/tex]:
The denominator can be factored using the difference of squares formula:
[tex]\[
x^4 - a^4 = (x - a)(x^3 + x^2a + xa^2 + a^3).
\][/tex]
### Step 2: Simplify the fraction
Substitute the factored forms into the original fraction:
[tex]\[
\frac{x^5 - a^5}{x^4 - a^4} = \frac{(x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4)}{(x - a)(x^3 + x^2a + xa^2 + a^3)}.
\][/tex]
Since [tex]\(x \neq a\)[/tex] (we are interested in the limit as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex], but [tex]\(x\)[/tex] is not equal to [tex]\(a\)[/tex] during this calculation), we can cancel the common factor [tex]\((x - a)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{x^4 + x^3a + x^2a^2 + xa^3 + a^4}{x^3 + x^2a + xa^2 + a^3}.
\][/tex]
### Step 3: Evaluate the limit as [tex]\(x \rightarrow a\)[/tex]
Now, we can substitute [tex]\(x = a\)[/tex] into the simplified expression:
[tex]\[
\lim_{x \to a} \frac{x^4 + x^3a + x^2a^2 + xa^3 + a^4}{x^3 + x^2a + xa^2 + a^3}.
\][/tex]
Replace [tex]\(x\)[/tex] with [tex]\(a\)[/tex]:
[tex]\[
\frac{a^4 + a^3a + a^2a^2 + aa^3 + a^4}{a^3 + a^2a + aa^2 + a^3}.
\][/tex]
Simplify the numerator and the denominator by combining like terms:
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]
Therefore, the limit becomes:
[tex]\[
\frac{5a^4}{4a^3} = \frac{5a}{4}.
\][/tex]
Thus, the limit is:
[tex]\[
\lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4} = \frac{5a}{4}.
\][/tex]
