To find the limit of the function [tex]\( f(x) = \frac{|x + 3|}{2x + 6} \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right ([tex]\(x \to -3^+\)[/tex]), we can proceed with the following steps:
1. Understand the function:
We have [tex]\( f(x) = \frac{|x + 3|}{2x + 6} \)[/tex]. The absolute value function [tex]\( |x + 3| \)[/tex] needs to be analyzed around the point where [tex]\( x \)[/tex] is approaching [tex]\(-3\)[/tex].
2. Behavior of the absolute value:
Since we are considering the limit as [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right ([tex]\(x \to -3^+\)[/tex]), the expression [tex]\( x + 3 \)[/tex] will be slightly positive. Therefore, [tex]\( |x + 3| = x + 3 \)[/tex] in this interval.
3. Simplify the function:
Since [tex]\( |x + 3| = x + 3 \)[/tex] when [tex]\( x \)[/tex] is slightly greater than [tex]\(-3\)[/tex], we can substitute this into the function:
[tex]\[
f(x) = \frac{x + 3}{2x + 6}
\][/tex]
4. Factor the denominator:
Notice that [tex]\( 2x + 6 \)[/tex] can be factored:
[tex]\[
2x + 6 = 2(x + 3)
\][/tex]
5. Simplify the fraction:
Substitute the factored form into the function:
[tex]\[
f(x) = \frac{x + 3}{2(x + 3)}
\][/tex]
When [tex]\( x \)[/tex] is approaching [tex]\(-3\)[/tex] from the right, [tex]\( x + 3 \)[/tex] is not zero and can be canceled out:
[tex]\[
f(x) = \frac{1}{2}
\][/tex]
6. Evaluate the limit:
As [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right, the function [tex]\( f(x) \)[/tex] becomes a constant:
[tex]\[
\lim_{x \to -3^+} \frac{|x + 3|}{2x + 6} = \frac{1}{2}
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \to -3^+} \frac{|x + 3|}{2x + 6} = \frac{1}{2}
\][/tex]
