Limits: Find the largest delta?

[mrjust]

Fist semester Calculus.

I'm working on problem #12 in the attachment. I'm supposed to find the largest possible delta, but I'm stuck.
So far I have the following:

lim (x->27) (x^(1/3))=3
epsilon=0.1
a=27
L=3
f(x)=(x^(1/3))

0< | x -a | < delta
| f(x) - L | < epsilon

0< | x -27 | < delta
| (x^(1/3)) - 3 | < epsilon

I know that I'm supposed to make epsilon look like delta, but I don't know how.

 

[DrPhil]

Fist semester Calculus.

I'm working on problem #12 in the attachment. I'm supposed to find the largest possible delta, but I'm stuck.
So far I have the following:

lim (x->27) (x^(1/3))=3
epsilon=0.1
a=27
L=3
f(x)=(x^(1/3))

0< | x -a | < delta
| f(x) - L | < epsilon

0< | x -27 | < delta
| (x^(1/3)) - 3 | < epsilon

I know that I'm supposed to make epsilon look like delta, but I don't know how.

How close to 27 does x have to be to guarantee that cbrt(x) is within 0.1 of 3?

Consider separately the cases for x>27 and x<27. Cube both sides of the inequality:

1) x>27, cbrt(x + delta) < 3.1 --> 27 + delta < 3.1^3 = 29.79 --> delta < 2.79

2) x<27, cbrt(x - delta) > 2.9 --> 27 - delta > 2.9^3 = 24.39 --> delta < 2.61

The stricter limit is delta < 2.61