The discussion revolves around finding the limits of the expression |x|/x as x approaches 0 from the positive and negative sides. Participants are exploring the behavior of the absolute value function and its implications for limits in calculus.
There is an active exploration of the concepts involved, with participants raising questions about the nature of absolute values and their effects on the limits. Some guidance has been provided regarding the approach to limits, emphasizing the importance of considering values approaching zero rather than substituting zero directly into the function.
Participants are navigating the definitions and implications of absolute values in the context of limits, with some expressing uncertainty about the process and reasoning behind their approaches.
dynamicsolo said:What does the absolute value operation do to positive numbers? to negative numbers?
flyingpig said:WHat is the definition of the absolute value!?
dynamicsolo said: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 said: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 said: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 said: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.