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If we graph the functions x2 and 2x, they intersect in three places:
One of the points of intersection has a negative value of x, and two have positive values of x. After those intersections, as x approaches ∞ the graph of 2x will always be on top of the function x2.
Here are some values of the two functions. We can see that when x = 2 and x = 4 the functions intersect, and that when x is greater than 4 the function 2x is pulling away from the function x2. It's like a horse race, where the function x2 is not having a good day.
If we take these values and look at the quotient , here's what happens:
We're already down to about 0.098 when x = 10, and we've barely started! When x = 100, we're down to
Now that's small. We conclude that
We already know these functions intersect for some negative value of x and at x = 2 and x = 4. We also know that the function x2 is losing the horse race: for all x greater than 4, x2 will be below (or behind) 2x.
Taking the same values, we'll look at the new ratios now that we've turned the fraction over.
These numbers don't seem to be approaching anything yet. Try something larger:
We find that as x gets larger, the quotient will also get larger and larger without bound. Therefore