To determine the limit [tex]\(\lim_{x \to -1} \frac{x^4 - 1}{x + 1}\)[/tex], we proceed with the following steps:
1. Analyze the expression:
The given limit is [tex]\(\lim_{x \to -1} \frac{x^4 - 1}{x + 1}\)[/tex]. Observe that substituting [tex]\(x = -1\)[/tex] directly into the expression yields a [tex]\(\frac{0}{0}\)[/tex] indeterminate form:
[tex]\[
\frac{(-1)^4 - 1}{-1 + 1} = \frac{1 - 1}{-1 + 1} = \frac{0}{0}
\][/tex]
2. Factor the numerator:
To resolve this indeterminate form, factor the numerator [tex]\(x^4 - 1\)[/tex]. Note that [tex]\(x^4 - 1\)[/tex] is a difference of squares:
[tex]\[
x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1)
\][/tex]
Further, [tex]\(x^2 - 1\)[/tex] is also a difference of squares:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
Putting everything together, we get:
[tex]\[
x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)
\][/tex]
3. Simplify the expression:
Substitute the factored form of [tex]\(x^4 - 1\)[/tex] back into the original limit expression:
[tex]\[
\lim_{x \to -1} \frac{(x - 1)(x + 1)(x^2 + 1)}{x + 1}
\][/tex]
Now, cancel the common factor [tex]\((x + 1)\)[/tex] in the numerator and the denominator:
[tex]\[
\lim_{x \to -1} (x - 1)(x^2 + 1)
\][/tex]
4. Evaluate the simplified limit:
After canceling, evaluate the limit of the remaining expression as [tex]\(x\)[/tex] approaches [tex]\(-1\)[/tex]:
[tex]\[
\lim_{x \to -1} (x - 1)(x^2 + 1)
\][/tex]
Substitute [tex]\(x = -1\)[/tex] into the simplified expression:
[tex]\[
(-1 - 1)((-1)^2 + 1) = (-2)(1 + 1) = (-2)(2) = -4
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \to -1} \frac{x^4 - 1}{x + 1} = -4
\][/tex]
