<p>Can someone explain how to do limits with absolute value terms? I’m confused on whether to use the positive or negative value of the term. Here is what I mean:</p>
<p>lim x->-3+ |x+3|/(x+3)
Is this… (x+3)/(x+3) OR -(x+3)/(x+3).</p>
<p>lim x->-3- |x+3|/(x+3)
Is this… (x+3)/(x+3) OR -(x+3)/(x+3).</p>
<p>lim x->-1- (x^2 +3x+2)/|x^2 -x-2|
Is this… (x^2 +3x+2)/(x^2 -x-2) OR (x^2 +3x+2)/-(x^2 -x-2)?</p>
<p>What is the rule that works EVERY TIME? How can you tell whether to use the positive or negative version of the absolute value sign?</p>
<p>It depends, but generally if the approach is from the left towards the cusp you use the negative value; from the right, the positive.
If evaluating limits not at the cusp (no matter what direction), use positive for anything to the right of the cusp and negative for anything to the left.</p>
<p>The most definitive way to get something that works each and every time is to try a value that’s relatively close to your value, and on each side.</p>
<p>So, for instance, in evaluating lim x->-3+ |x+3|/(x+3), you want to substitute a value that’s a little bit larger than -3, such as -2.99. When you substitute -2.99, x+3 = 0.01 > 0, so |x+3| = x+3.</p>
<p>If you were looking to evaluate lim x->-3- |x+3|/(x+3), you now want to substitute a value that’s a little bit less than -3, such as -3.01. When you substitute -3.01, x+3 = -0.01 < 0 so |x+3| = -(x+3).</p>
<p>The reason why you want to avoid thinking like “Are you looking from the right hand side? Then it’s positive…” is for examples like lim x->3+ |3-x|/(3-x). Here, that rule fails. However, when you pick a number slightly larger than 3, such as 3.01, and plug it in, 3 - x = -0.01 < 0, so |3-x| = -(3-x).</p>
<p>Notice that this same piece of advice will serve you well when not evaluating at values that yield 0/0.</p>
<p>Hope that helps.</p>
<p>Thanks. I understand it now. I use to think that if it approaches from the left, it’s automatically negative, and vice-versa.</p>