You know that for any you can find an so that . In particular, this says that for any you can find an x so that . Can you see why that is? Can you then conclude anything about your limit?
Let's say that for all (we know by definition that we can find such an N for each ). From to we're changing the input by x (2x-x=x). Since the absolute value of the derivative is always less than epsilon in this interval, we can definately not move further from f(x) than when we move x units ahead. Therefore .
Let's say that for all (we know by definition that we can find such an N for each ). From to we're changing the input by x (2x-x=x). Since the absolute value of the derivative is always less than epsilon in this interval, we can definately not move further from f(x) than when we move x units ahead. Therefore .