To evaluate the limit [tex]\(\lim_{{b \to -3^{-}}} \frac{|b+3|}{b+3}\)[/tex], let's go through it step by step.
1. Understand the expression inside the limit:
We have the expression [tex]\(\frac{|b+3|}{b+3}\)[/tex]. The absolute value function [tex]\(|b+3|\)[/tex] will always yield a non-negative result. However, [tex]\(b+3\)[/tex] itself can be either positive or negative depending on the value of [tex]\(b\)[/tex]:
- When [tex]\(b > -3\)[/tex], [tex]\(b+3\)[/tex] is positive, so [tex]\(|b+3| = b + 3\)[/tex].
- When [tex]\(b < -3\)[/tex], [tex]\(b+3\)[/tex] is negative, so [tex]\(|b+3| = -(b + 3)\)[/tex], which equals [tex]\(-(b + 3)\)[/tex].
2. Evaluate the behavior as [tex]\(b \to -3\)[/tex] from the left ([tex]\(b \to -3^-\)[/tex]):
When [tex]\(b\)[/tex] approaches [tex]\(-3\)[/tex] from the left, it means [tex]\(b < -3\)[/tex].
For such values:
[tex]\[
\frac{|b+3|}{b+3} = \frac{-(b+3)}{b+3}
\][/tex]
3. Simplify the expression:
When [tex]\(b < -3\)[/tex]:
[tex]\[
\frac{|b+3|}{b+3} = \frac{-(b+3)}{b+3} = \frac{-1 \cdot (b+3)}{b+3}
\][/tex]
Since [tex]\(b+3 \neq 0\)[/tex] (as [tex]\(b \neq -3\)[/tex]):
[tex]\[
\frac{|b+3|}{b+3} = -1
\][/tex]
4. Conclusion:
As [tex]\(b\)[/tex] approaches [tex]\(-3\)[/tex] from the left ([tex]\(b \to -3^{-}\)[/tex]), the value of [tex]\(\frac{|b+3|}{b+3}\)[/tex] simplifies to [tex]\(-1\)[/tex].
Therefore, we have:
[tex]\[
\lim_{{b \to -3^{-}}} \frac{|b+3|}{b+3} = -1
\][/tex]
This completes our step-by-step evaluation of the limit. The answer is [tex]\(-1\)[/tex].
