Question

calculate the limit \\(\lim_{x \to a} \frac{x^5 - a^5}{x - a}\\) using the following factorization formula where n is a positive integer and a is a real number.

\\x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \dots + xa^{n-2} + a^{n-1})\\

\\(\lim_{x \to a} \frac{x^5 - a^5}{x - a} = \square\\)

Explanation:

Apply the factorization formula for \(n = 5\)

$$ x^5 - a^5 = (x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4) $$

Simplify the rational expression

$$ \frac{x^5 - a^5}{x - a} = x^4 + x^3a + x^2a^2 + xa^3 + a^4 \quad \text{for } x eq a $$

Evaluate the limit by direct substitution

$$ \lim_{x \to a} (x^4 + x^3a + x^2a^2 + xa^3 + a^4) = a^4 + a^4 + a^4 + a^4 + a^4 = 5a^4 $$

Answer:

Calculate the limit \(\lim_{x \to a} \frac{x^5 - a^5}{x - a}\) using the following factorization formula where n is a positive integer and a is a real number.

$$x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \dots + xa^{n-2} + a^{n-1})$$

\(\lim_{x \to a} \frac{x^5 - a^5}{x - a} =\) <blank>\(5a^4\)</blank>