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 [itex]\frac{|x|}{x} [/itex] equal?
If x is any negative number, what will | x | equal? And now what will [itex]\frac{|x|}{x} [/itex] 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.
A limit is the value that a function approaches as its input approaches a certain point or value. It is not necessarily the exact value of the function at that point, but rather the value that the function gets closer and closer to as the input gets closer to the specified point.
The limit of |x|/x at x=0+ is undefined. This is because the function approaches positive infinity as the input approaches 0 from the right side, and negative infinity as the input approaches 0 from the left side. Since these values are not equal, the limit does not exist.
The limit of |x|/x at x=0- is also undefined. This is because the function approaches positive infinity as the input approaches 0 from the left side, and negative infinity as the input approaches 0 from the right side. Again, since these values are not equal, the limit does not exist.
The limit of |x|/x does not exist at x=0 because the function approaches different values from the left and right sides of 0. The left and right limits do not equal each other, so the overall limit does not exist.
No, the limit of |x|/x cannot be defined at x=0 because the function has different left and right limits at this point. In order for a limit to exist, the left and right limits must be equal.