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Unformatted text preview: MAT 137Y, Limit Proofs and Some Examples We will do our first example at great length with much commentary and explanation. You should not regard it as a model for your proofs but as a guide to thinking through your solution. Example 1. Find a number δ > 0 such that | x- 2 | < δ implies | x 3- 8 | < 10- 4 . In other words find a δ > such that whenever x satisfies | x- 2 | < δ then it also satisfies | x 3- 8 | < 10- 4 . First Solution. We have to make | x 3- 8 | small (specifically, less than 10- 4 ) by making x “sufficiently close” to 2, namely by making | x- 2 | “sufficiently small”. Imagine that someone (Peggy, say) has challenged us to make | x 3- 8 | small and we get to decide how small we want to make | x- 2 | in order to achieve this. The smallness of | x- 2 | is controlled by the number δ which we are going to choose. We are going to choose this δ and then prove to Peggy that it works, that is that | x- 2 | < δ implies that | x 3- 8 | < 10- 4 . (To put it another way we are going to show her that no matter what value of x she cares to take in the interval ( 2- δ , 2 + δ ) then, for that value of x , f ( x ) will lie in the interval ( 8- 10- 4 , 8 + 10- 4 ) . It is essential to understand that there is no one “correct” value for δ . True, there is theoretically a largest possible value of δ which will work but it will generally be difficult if not impossible to find. Fortunately we don’t have to find it! We are quite free to make δ much smaller than it really needs to be. It may well be that the δ we provide to Peggy in fact makes x 3- 8 < 10- 10 . If so, so much the better: overkill is certainly not forbidden, and in fact makes life easier. Speaking of overkill it is also worth noting that if we find a value δ which works then any smaller value of δ , for example δ / 2, will also work (why?). Having made these preliminary comments let’s get to work. What we are going to do is work with the expression | x 3- 8 | to see how we can make it less than 10- 4 . Our work will basically be a long chain of inequalities | x 3- 8 | < ...... < 10- 4 . Some of these strict inequalities may be replaced by “less than or equal to” or “equal to”: as long as we have at least one strict inequality in our chain we can conclude that the first element of the chain is less than the last. At each step we will have to justify the inequality. Sometimes the inequality will hold without any assumptions on x (for example if we apply the triangle inequality). Sometimes the inequality will follow from a specific assumption we make about how small | x- 2 | is ( for example | x- 2 | < 1 25 or | x- 2 | < 1). We are free to make as many such assumptions as we like since the smallness of | x- 2 | is completely under our control....
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- Spring '10
- lim, |X