Transcription of Squeeze Theorem Examples - Grove City College
1 Squeeze Theorem ExamplesSqueeze (x) g(x) h(x)whenxis neara(but not necessarily ata[for instance, g(a) may be undefined]) andlimx af(x) = limx ah(x) =L,thenlimx ag(x) = 0x2cos(1x2).When trying to find functions to use to Squeeze g(x), we want functions that are, a) similar enough tog(x) that wecan be sure the Squeeze works, b) easier to evaluate their limit asx a. We typically do this by starting with the mostcomplicated or troublesome part ofg(x), see if we can find constants (or simpler functions) that it stays between, andthen multiply in the rest of nicer parts ofg(x).In this case, the part ofg(x) that is giving us the most trouble is the cos(1x2) part (we get division by 0 if we try directsubstitution). Now we know that cosine stays between -1 and 1, so 1 cos(1x2) 1for anyxin the domain of the function ( , anyx6= 0). Sincex2is always positive, we can multiply this inequalitythrough byx2: x2 x2cos(1x2) x2So, our original function is bounded by x2andx2.
2 Now sincelimx 0 x2= limx 0x2= 0,then, by the Squeeze Theorem ,limx 0x2cos(1x2)= 0x2esin(1x).As in the last example , the issue comes from the division by 0 in the trig term. Now the range of sine is also [ 1,1], so 1 sin(1x) to both sides of an inequality does not change the inequality, soe 1 esin(1x) e1,1and once again we can multiply through byx2and getx2e 1 x2esin(1x) , our original function is bounded bye 1x2andex2, and sincelimx 0e 1x2= limx 0ex2= 0,then, by the Squeeze Theorem ,limx 0x2esin(1x)=
