Example 2.Show that limx→∞1x2= 0.Solution.Fixε >0.We needcso that ifx > cthen 1/x2< ε.Choosingc= 1/√εworks.Basic limit lawsOne can show that the basic limit laws hold for limits at±∞.So if limx→∞f(x) =A,limx→∞g(x) =BwhereA, Bare real numbers (i.e. not infinite!) thenf(x) +g(x), f(x)-g(x), f(x)g(x) will also have limits at∞(A+B, A-BandAB), and ifB6= 0 thenf(x)/g(x)will have a limitA/B. (Similar statements hold at-∞.) You can also use the SqueezingPrinciple, this is especially useful if you can estimate complicated functions with simplerones.If you want to use the limit laws for infinite limits then you have to be a bit more careful:remember that∞and-∞are not real numbers. However you can still ‘pretend’ that certainoperations can be carried out:‘∞+∞=∞’, ‘∞ × ∞=∞’, ‘c× ∞=∞’ ifc >0 and ‘c× ∞=-∞’ ifc <0, ‘c∞= 0’,c+∞=∞’ etc. You should actually use these as rules for limits, not as the outcomes ofoperations on∞. That means that you will have statements like this:Assume thatlimx→pf(x) =∞andlimx→pg(x) =∞.Then•limx→p(f(x) +g(x)) =∞1
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•limx→pf(x)g(x) =∞•limx→p4f(x)= 0.Note that iffandghave∞limits atpthen we cannot say anything about the limit off(x)-g(x) andf(x)/g(x) there. (It can be basically anything or it might not exist.)Polynomials and rational functionsIt is not hard to show that for any non-constant polynomialp(x) the limit ofp(x) will be∞or-∞adx
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