Get expert advice and community support for all your questions on IDNLearn.com. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
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].
[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].
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.