Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

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].
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.