# Delta-epsilons problem

1. Sep 5, 2010

### Shaidester

1. The problem statement, all variables and given/known data

1) What is the limit of f(x) = 3x2 - 2x - 4 as x approaches 2?

Obviously it is 4, now to 1a)

1a) For epsilon = 0.01, find a value of delta so that f(x) is closer to the limit than epsilon. Apply the same for when epsilon = 0.001.

2) What is the limit of f(x) = sqrt(x) as x approaches 9?

Obviously it is 3, now to 2a)

2a) For epsilon = 0.01, find a value of delta so that f(x) is closer to the limit than epsilon. Apply the same for when epsilon = 0.001.

2. Relevant equations

I know how to prove the limit of linear functions but I am not sure about square root and quadratic functions. My teacher has only covered one problem and it was the last ten minutes of class, so it was rushed and not clearly explained.

3. The attempt at a solution

1a) Let epsilon = 0.01

•|f(x) - L| < epsilon

•|3x^2 - 2x - 4 - 4| < 0.01

•-0.01 < 3x^2 - 2x - 8 < 0.01

Now, where do I go from here?

2a) Let epsilon = 0.01

•|f(x) - L| < epsilon

•|sqrt(x) - 3| < 0.01

•-0.01 < sqrt(x) - 3 < 0.01

•2.99 < sqrt(x) < 3.01

•2.992 < sqrt(x) < 3.012

•8.9401 < x < 9.0601

Now, where do I go from here?

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