Math 231E. Fall 2013. Lecture 3B.Limits at Infinity.1 Limits at ∞ and horizontal asymptotesWe now would like to define limits as x → ∞, and we can do so in the following manner:We say thatlimx→∞f(x) = L, if limx→0+f1x= L.In short, we first plug in 1/x for x, then take the limit x → 0 from the right.Example 10. We want to computelimx→∞3x2+ 2x − 7x2− 4x + 1.We plug in 1/x for x to obtain3(1/x)2+ 2(1/x) − 7(1/x)2− 4(1/x) + 1=3x2+2x− 71x2−4x+ 1=1/x21/x2·3 + x − 7x21 − 4x + x2,and taking the limit x → 0+, we obtain 3.We also say thatlimx→−∞f(x) = L, if limx→0−f1x= L.Example 11. Let us computelimx→∞exxn,for some fixed n > 0. We know that ex> xn+1/(n + 1)!. (Why is this?) Then we haveexxn>xn+1xn(n + 1)!=x(n + 1)!.Now,limx→∞x(n + 1)!= limx→0+1/x(n + 1)!=1(n + 1)!limx→0+1x= ∞.Thereforelimx→∞exxn= ∞.Example 12. Similarly, we would like to computelimx→∞xne−x.From the arguments above, we have thatxnex<(n + 1)!x.Also, for x > 0, we have xne−x> 0, so0 <xnex<(n + 1)!x.We computelimx→∞(n + 1)!x= limx→0+(n + 1)!1/x= (n + 1)! limx→0+x = 0,so by the Squeeze Theorem we havelimx→∞xne−x= 0.This is why we would say that “e−xdecays to zero faster than any polynomial as x → ∞”.2 Asymptotes and graphingSee
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