To find the limit of the expression
[tex]\[
\lim_{k \to -2} \frac{k^3 + 8}{k^2 - 4},
\][/tex]
let's work through the problem step-by-step:
1. Factorize the numerator and denominator:
The numerator [tex]\( k^3 + 8 \)[/tex] can be factorized using the sum of cubes formula, which states that [tex]\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)[/tex]. Here, [tex]\( a = k \)[/tex] and [tex]\( b = 2 \)[/tex], so:
[tex]\[
k^3 + 8 = k^3 + 2^3 = (k + 2)(k^2 - 2k + 4).
\][/tex]
The denominator [tex]\( k^2 - 4 \)[/tex] can be factorized using the difference of squares formula, which states that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex]. Here, [tex]\( a = k \)[/tex] and [tex]\( b = 2 \)[/tex], so:
[tex]\[
k^2 - 4 = (k - 2)(k + 2).
\][/tex]
2. Rewrite the limit expression with the factored forms:
Substitute the factored forms into the original expression:
[tex]\[
\frac{k^3 + 8}{k^2 - 4} = \frac{(k + 2)(k^2 - 2k + 4)}{(k - 2)(k + 2)}.
\][/tex]
3. Simplify the expression by canceling common factors:
We can cancel the common factor [tex]\( k + 2 \)[/tex] in the numerator and the denominator:
[tex]\[
\frac{(k + 2)(k^2 - 2k + 4)}{(k - 2)(k + 2)} = \frac{k^2 - 2k + 4}{k - 2}.
\][/tex]
4. Evaluate the limit of the simplified expression:
Now, we consider the limit of the simplified expression as [tex]\( k \)[/tex] approaches [tex]\(-2\)[/tex]:
[tex]\[
\lim_{k \to -2} \frac{k^2 - 2k + 4}{k - 2}.
\][/tex]
Substitute [tex]\( k = -2 \)[/tex] into the simplified expression:
[tex]\[
\frac{(-2)^2 - 2(-2) + 4}{-2 - 2} = \frac{4 + 4 + 4}{-4} = \frac{12}{-4} = -3.
\][/tex]
So, the limit is:
[tex]\[
\lim_{k \to -2} \frac{k^3 + 8}{k^2 - 4} = -3.
\][/tex]
