Discover a wealth of knowledge and get your questions answered at IDNLearn.com. Find in-depth and trustworthy answers to all your questions from our experienced community members.

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].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.