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Evaluate the limit:

[tex]\[ \lim_{x \rightarrow 2} \frac{x^2 + 6x + 8}{x + 4} \][/tex]


Sagot :

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].