How to Show f(x) is Close to L Using Epsilon-Delta Proof

[John O' Meara]

Generate the graph of f(x)=x^3-4x+5 with a graphing utility. And use the graph to find delta such that |f(x)-2|<.05 if 0<|x-1|<delta [Hint show that the inequality |f(x)-2| < .05 can be rewritten 1.95 < x^3-4x+5 < 2.05, then estimate the values of x for which f(x)=1.95 and f(x)=2.05]. Now I can do the graphing parts and I can correctly estimate the values of x, but I cannot do the "show that the inequality |f(x)-2|<.05 can be rewritten etc." Can anyone show me how to do it as I have started studying a University maths book on my own. Thanks.

 

[snipez90]

|f(x) - L| < e means that f(x) is in (L - e, L + e) which means that L - e < f(x) < L + e. Also, I'm not sure why the text asks you to use a graphing utility since it's maybe instructive once or twice to see an epsilon-delta proof done with an explicit (numerical) epsilon to get a feel what is meant by "closeness" of f(x) to L or whatever.