Find Limits of |x|/x at x=0+ and x=0-

[Nitrate]

Homework Statement

Find the following limits, if they exist:

h) lim absolute value[x]/(x)
x->0^{+}

and

i) lim absolute value[x]/(x)
x->0^{-}

Homework Equations

Not sure

The Attempt at a Solution

The answer to h) [according to the back of the textbook] is 1
and the answer to i) is -1.
I'm really not sure how to arrive at those answers.

 

[dynamicsolo]

What does the absolute value operation do to positive numbers? to negative numbers?

 

[flyingpig]

WHat is the definition of the absolute value!?

 

[Nitrate]

What does the absolute value operation do to positive numbers? to negative numbers?

Positives stay the same and negatives turn to positives, but I'm not sure how this will help me.

 

[dynamicsolo]

So if x is any positive number, what will | x | equal? What will \frac{|x|}{x} equal?

If x is any negative number, what will | x | equal? And now what will \frac{|x|}{x} equal?

 

[Nitrate]

WHat is the definition of the absolute value!?

The distance of x from 0.

 

[Nitrate]

So if x is any positive number, what will | x | equal? What will \frac{|x|}{x} equal?

If x is any negative number, what will | x | equal? And now what will \frac{|x|}{x} equal?

I see it now, but why can x be ANY positive number for the first question and ANY negative number for the second? I thought you had to substitute the zero into the x variable.

 

[dynamicsolo]

I see it now, but why can x be ANY positive number for the first question and ANY negative number for the second? I thought you had to substitute the zero into the x variable.

Because we are not just plugging numbers into the function; we are looking to see what happens as x takes on (any) positive or negative values that are getting closer and closer to zero. (This is important because it is often the case in limit problems (like this one) where putting the "limiting value of x" into the function won't tell you anything useful at all...)

 

[Nitrate]

Because we are not just plugging numbers into the function; we are looking to see what happens as x takes on (any) positive or negative values that are getting closer and closer to zero. (This is important because it is often the case in limit problems (like this one) where putting the "limiting value of x" into the function won't tell you anything useful at all...)

How can I tell what the graph of abs(x)/(x) looks like? [I've already used my graphing calculator, but I want to find out how to do it without my calculator] I know that the abs(x) by itself looks like a v opening up.

 

[dynamicsolo]

How can I tell what the graph of abs(x)/(x) looks like? [I've already used my graphing calculator, but I want to find out how to do it without my calculator] I know that the abs(x) by itself looks like a v opening up.

All right, what sort of graph do you get if you take the values of y = abs(x) and divide them by the value of x at each point?