Question

calculate the following limit using the factorization formula $x^{n}-a^{n}=(x - a)(x^{n - 1}+x^{n - 2}a+x^{n - 3}a^{2}+cdots+xa^{n - 2}+a^{n - 1})$ where $n$ is a positive integer and $a$ is a real number. $limlimits_{x
ightarrow3}\frac{x^{4}-81}{x - 3}$ (simplify your answer.)

Explanation:

Step1: Factorize $x^{4}-81$

Using the formula $x^{n}-a^{n}=(x - a)(x^{n - 1}+x^{n - 2}a+\cdots+xa^{n - 2}+a^{n - 1})$, when $n = 4$ and $a = 3$, we have $x^{4}-81=(x - 3)(x^{3}+3x^{2}+9x + 27)$.

Step2: Substitute into the limit

$\lim_{x
ightarrow3}\frac{x^{4}-81}{x - 3}=\lim_{x
ightarrow3}\frac{(x - 3)(x^{3}+3x^{2}+9x + 27)}{x - 3}$.

Step3: Cancel out the common factor

Cancel out the $(x - 3)$ terms in the numerator and denominator, we get $\lim_{x
ightarrow3}(x^{3}+3x^{2}+9x + 27)$.

Step4: Evaluate the limit

Substitute $x = 3$ into $x^{3}+3x^{2}+9x + 27$. Then $3^{3}+3\times3^{2}+9\times3+27=27+27 + 27+27=108$.

Answer:

$108$