I was just introduced to the http://en.wikipedia.org/wiki/Squeeze_theorem" [Broken] in the second week of Calc 1. This theorem implies that [itex]\lim_{x \to c} f(x)[/itex] exists under certain circumstances, and I want to find out and prove what (at least some of) those circumstances are and what the value of the limit is.(adsbygoogle = window.adsbygoogle || []).push({});

Roughly, my idea so far is to use Squeeze and supremums and infimums to find the one-sided limits, giving me the two-sided limit when they agree. I would like to know if I am messing up somewhere or if anyone has some time-saving hints for me. (This is not homework, so I don't have much time to spend on it.) All functions are real-valued.

Theorem X, rough draft.I need to cover 4 cases: increasing from the left, decreasing from the left, increasing from the right, and decreasing from the right.

(a) If f is bounded and increasing (or decreasing) on [b,c), then [itex]\lim_{x \uparrow c} f(x)[/itex] = the supremum (or infimum) of the image of f on [b,c).

(b) If f is bounded and increasing (or decreasing) on (c,d], then [itex]\lim_{x \downarrow c} f(x)[/itex] = the infimum (or supremum) of the image of f on (c,d].

PROOF

(a-increasing) Let s = the supremum of the image of f on [b,c). The pair of squeezing functions will be g(x) = s and h(x) = ((s - f(b))/(c - b))x, i.e., a horizontal line through s and a secant line to f passing through f(b) and s. That these two functions will squeeze f looks obvious since f is increasing on the interval. And the supremum should exist since I am dealing with subsets of R.

I also need to prove that the limits of my squeezing functions g(x) and h(x) always exist and are equal approaching any c. Both limits exist since g(x) is constant and h(x) is linear and defined everywhere since c > b. Should I note that c > b or is this assumed for an interval? I'm not sure how to prove that the limits are equal.

The other three cases would work similarly.

My last step is applying Squeeze. Is there a one-sided version of Squeeze? if not, I would have to combine the g(x) and h(x) that I get from the left with the ones that I get from the right. If the one-sided limits are equal, g should be the same horizontal line through s, and h should be either a line through s or an absolute value function with vertex s. So combining them should be straightforward.

Will this work? Will f need to be continuous on the interval?

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# Implications of Squeeze Theorem

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