Let's solve the limit:
[tex]\[
\operatorname{Lim}_{x \rightarrow 2} \frac{x^2 + 6x + 8}{x + 4}
\][/tex]
Step-by-step:
1. Understand the expression: The given expression is a rational function [tex]\(\frac{x^2 + 6x + 8}{x + 4}\)[/tex].
2. Factor the numerator if possible: We notice that [tex]\(x^2 + 6x + 8\)[/tex] can be factored. Let's factorize it.
[tex]\[
x^2 + 6x + 8 = (x + 2)(x + 4)
\][/tex]
3. Rewrite the expression: Substituting the factorized form back into the numerator:
[tex]\[
\frac{(x + 2)(x + 4)}{x + 4}
\][/tex]
4. Simplify the expression: Now, if [tex]\(x \neq -4\)[/tex] (which is outside the domain of our interest since we are looking at [tex]\(x\)[/tex] approaching 2), we can cancel the [tex]\((x + 4)\)[/tex] terms.
[tex]\[
\frac{(x + 2)(x + 4)}{x + 4} = x + 2 \quad \text{(for } x \neq -4)
\][/tex]
5. Substitute [tex]\(x = 2\)[/tex] in the simplified expression: Now that we have simplified the function, we can directly substitute [tex]\(x = 2\)[/tex] into the simplified expression [tex]\(x + 2\)[/tex].
[tex]\[
x + 2 \quad \text{at} \quad x = 2
\][/tex]
[tex]\[
2 + 2 = 4
\][/tex]
6. Conclusion: Therefore, the limit of the given function as [tex]\(x\)[/tex] approaches 2 is:
[tex]\[
\operatorname{Lim}_{x \rightarrow 2} \frac{x^2 + 6x + 8}{x + 4} = 4
\][/tex]
Thus, the value of the limit is [tex]\(\boxed{4}\)[/tex].
