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Themulandeightheerenm Cor Squeeze Theorem Here is a really tricky limit 41 X's in f How do we calculate it The function y xcos I is not continuous at x O sowecan't just plug in X 0 Instead recall that I E sin 101E l for all oc IR Take O T I E Sin H e l Now multiply through by x2 x 2 e x 2 sin Ix E XZ A Cx B Note that the E does not change directionwhen multiplying by x2 because x2 o This shows that our function is in Hx is sandwiched between the two functions A X XZand BCN XZ whose limits are diff A Cx o no Ban But since it sinks is sandwiched in between these two limits the only possibility is a info x2 sin xt O This strategy of sandwiching a function to find a limitis called the sandwichtheorem BCH f A 11 SANDWICHTH EOREM LetAxl be a function and let ac IR Suppose that Axl Ballare two functionssuch that Axl E f CH E BHI for all near a and fifa Hx ma BAI L Then finna fail L This theorem is most useful when dealingwith sine and cosine limits Cos Exemple Let's evaluate 7 Starting from l eSinan e 1 and multiplyingthrough by Hx noting that xco since x is going to x we get theinequalities if 3 0 13 a the inequality got reversed because Ko Since the top and bottom of the sandwich namely and I go to Zero as x no it follows from the Sandwich Theorem that III 05 o
