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

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  • Thread starter Thread starter John O' Meara
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SUMMARY

The discussion focuses on demonstrating that the function f(x) = x³ - 4x + 5 is close to the limit L = 2 using an epsilon-delta proof. Participants emphasize rewriting the inequality |f(x) - 2| < 0.05 as 1.95 < f(x) < 2.05 to find the appropriate delta. The graphing utility is recommended for visualizing the function's behavior around x = 1, aiding in estimating the values of x that satisfy the conditions. The conversation highlights the importance of understanding the epsilon-delta definition of limits in calculus.

PREREQUISITES
  • Understanding of epsilon-delta definitions in calculus
  • Familiarity with polynomial functions and their properties
  • Experience using graphing utilities for function analysis
  • Basic knowledge of inequalities and their manipulation
NEXT STEPS
  • Learn how to apply epsilon-delta proofs in various limit scenarios
  • Explore graphing techniques using tools like Desmos or GeoGebra
  • Study polynomial behavior and its implications for limits
  • Practice rewriting inequalities to facilitate limit proofs
USEFUL FOR

Students studying calculus, particularly those learning about limits and epsilon-delta proofs, as well as educators seeking to enhance their teaching methods in mathematical analysis.

John O' Meara
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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.
 
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|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.
 

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