Expert: Paul Klarreich - 9/16/2007

Question
The two problems are the following:

Give the proof of a limit for a linear function,and the proof of the sum and product of special functions.

I'm guessing you have to use the delta-epsilon definition, but am not sure how to go about it.  Thanks.

Answer
Questioner:   Len
Category:  Calculus
Private:  No
 
Subject:  calculus: limit proofs
Question:  The two problems are the following:

Give the proof of a limit for a linear function,and the proof of the sum and product of special functions.

I'm guessing you have to use the delta-epsilon definition, but am not sure how to go about it.  Thanks.
........................................................
Hi, Len,

I am afraid your question is not at all clear.

I suspect your first one says:

Prove that  

lim   ax + b = ar + b
x->r

but I am not sure what you mean by 'special functions.'  In the meantime, we can give the first one a try.  You have to prove:  [I cannot make epsilons and deltas, so I will use e and d.]

Given  e > 0, there exists  d > 0  such that:

whenever | x - r | < d,  |ax + b - (ar + b) | < e

Now |ax + b - (ar + b) | =

|ax + b - ar - b | =

|ax - ar | =

|a(x - r)| =

|a| |x - r|

So if |x - r| < d then

|a||x - r | < d|a|

So all we need is to take  d = e/|a| and we have our proof.