Thread: Precise Def of a Limit Problem

Given the limit as x approaches 1 of (4 + x -3x^3) = 2 find the value of delta that correspond to sigma = 0.25.

I never got to these with my self-study. Can someone explain this type of problem quickly? I understand the concept but I don't know how to apply it.

I know it gets started like 4 + x - 3x^3 < .25 or something. But I don't know what to do after that

d=delta
sigma=e


|x-1|<d ==> |4+x-3x^3-2|<1/4

|2+x-3x^3|=|(x-1)(3x^2+3x+2|<d|3x^2+3x+2|

|x-1|<d ==>x<d+1 , |3x^2+3x+2|<3(d+1)^2+3(d+1)+2=3d^2+6d+3+3d+3+2=3d^ 2+9d+8

d|3x^2+3x+2|<d(3d^2+9d+8)<1/4

==> 3d^3+9d^2+8d-1/4<0

==> d<-1+1/36 (17496-216 sqrt(6497))^(1/3)+1/6 (81+sqrt(6497))^(1/3)~2.03

Note that there is no such thing as the delta that corresponds to a particular epsilon. If one value of delta works, then so does any smaller value.