prove that the lim as x goes to 4 of x^2 + x -11 = 9
This is the example used on Paul's Online Notes on limits in calculus which can be found here http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx (I really like this resource.)
Paul factors x^2 + x - 11 to get /x+5//x-4/ < Epsilon, then notes that if we can find a value K that is greater than /x+5/ we can say /x+5//x-4/ < K/x-4/
(I am using / for absolute value bars, is there a better convention?)
From there he makes the assumption that K/x-4/ < Epsilon
How does he make that assumption?
The Attempt at a Solution
To me it looks like this:
/x+5//x-4/ < Epsilon and /x+5//x-4/ < K/x-4/
Therefor Epsilon = K/x-4/ and /x-4/ = Epsilon/K
Which would be great except that if that is true, the rest of the proof doesn't work. What am I missing about the relationship of the inequalities?