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Use the Squeeze Theorem to show that $ \displaystyle \lim_{x \to 0}(x^2 \cos 20\pi x) = 0 $. Illustrate by graphing the functions $ f(x) = -x^2 $, $ g(x) = x^2 \cos 20\pi x $, and $ h(x) = x^2 $ on the same screen.

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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Idaho State University

this problem. Number thirty five of the sewer Calculus Expedition Section two point three Use a squeezed serum to show that element is export zero of the function X squared times co sign of the quantity twenty by X equals zero and we're going to illustrate to squeeze serum by graphing the functions and f of X is equal to negative. X squared g m x is equal to X squared cosa in twenty pi x and h of X is equal to x squared on the same the screen. So we begin with a squeeze term, Um, which states that if you have two functions f n g, whose limits as ex purchase some value, eh is equal to value? No. So for a friend Jean, this limited equivalent and no his list them are equal to let me replace ms with each h b in the upper limit upper function jean de target function. And if this is true, then a limit as expressions, eh? For Jean, it's also equal to hell, so we just need to prove them Disfunction Gene is indeed between these two functions and then also confirmed that those two limits are equivalent so first, Maybe in with I'm focusing on the coast. Ain't part of this function. Really Recall that the Cho sang graph for the crossing function oscillates detain the values of a negative one and positive one. I simply want playing each term here by X squared, which is always a positive value. We see that we can now compare the three functions in question. Here we have f arrivederci area each and we have confirmed that if his less than Jean listening to Gene and the G is less than or equal to H now we want to confirm what is the limit and six approaches zero of F or negative X squared. And we see here clearly this limits equal zero. How about for H? Tell him his expert zero h which x squared, you see, that's also equal to zero. And because this is true And this is true by the squeeze Dirham, the limit and six purchase zero uh, X squared Hussein twenty pie. Thanks. It's absolutely out of there. We have confirmed that algebraic lee let us show graphically how this is true. Here is a plot of all three functions on the sing graph The minimum plant negative. X squared is at the bottom by the Orange Point, which is our target function. And then the Purple Plan X squared, which is eight tracks the upper function. Even though thie middle function his larger than X squared and Liston negative x squared at some point for the limit as expert to zero. So this area, specifically we see that it is most definitely between made of X squared X squared. And we see that telemetry a criminal of inventory indeed zero and that confirms that were a solution here with disgrace there.