Limits Involving Absolute Value: Solutions & Explanations

[The Chaz]

Let f(x) = |x|/x

a. What is the limit of f, as x approaches 0 from the right?
b. What is the limit of f, as x approaches 0 from the left?
c. Hence, what is the limit of f, as x approaches 0?
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The best way to evaluate limits involving absolute values is to use the definition of absolute value

When x > 0, the absolute value of x is just x (by definition). We write |x| = x.
When x < 0, the absolute value of x is the opposite of x. We write |x| = -x.

a. As x approaches 0 from the right, x is positive (x > 0). So we can replace |x| with x to write..
(The limit of x/x, as x approaches 0 from the right) = (The limit of 1 ...) = 1.

b. Likewise, we replace |x| with -x and arrive at -1.

c. Does not exist

(I needed a few more posts, but didn't want to just fill the forum with garbage. In the future, I'll expand this to be a more general discussion of limits involving absolute values, and maybe it would be sticky-worthy...)

 

[Markov2]

Re: Limits involving aboslute value - here is a solution to a commonly asked question.

Good call, perhaps we could produce a PDF with very commonly asked questions!