Prove the statement using the epsilon-delta definition of limit: lim (x?4) x^2 - 2x + 1 = 9

Question

Prove the statement using the epsilon-delta definition of limit: lim (x→4) x² - 2x + 1 = 9

 

#21, practice test

Answer 1

 

Hi Mia,

 

You can begin with ε > 0, and find δ > 0 (which depends on ε)

so that if 0 < |x - 4| < δ, it follows that |f(x) - 9| < ε.

 

Beginning with

     |(x² - 2x + 1) - 9| < ε

 

     |x² - 2x -8| < ε

 

     |x + 2||x - 4| < ε        (Eq. 1)

 

If we arbitrarily assume δ ≤ 1,

 

     |x - 4| < δ ≤ 1

 

     |x - 4| < 1

 

     3 < x < 5

 

from which we can derive

 

     5 < |x + 2| < 7           (Eq. 2)

 

From Eq. 1 and Eq. 2,

 

     |x + 2||x - 4| < 7|x - 4| < ε

 

     |x - 4| < ε/7

 

Let's set δ = min{1, ε/7}.

 

So, if 0 < |x - 4| < δ, it follows that |f(x) - 9| < ε.