The document provides a formal proof using the epsilon-delta definition of limits to show that as x approaches 7, 10x approaches 70. It demonstrates that by picking δ = ε/10, the condition |10x - 70| < ε can be satisfied. Additionally, the document encourages exploring variations of the proof with different limits and functions.

1. Limit Proofs
A Resource for WGU Calculus III
By Gideon Weinstein

2. The Problem
Use the formal epsilon-delta definition of limit to
prove the following:
lim 10 x 70
x 7

3. The Proof
By the formal definition, lim f ( x ) L
x a
means that for any arbitrary ε>0, there
exists a δ>0 so that the following is
true:
when x is chosen so |x-a| < δ
then |f(x)-L| < ε

4. The Strategy
We need to SHOW that lim 10 x 70
x 7
which means that we need to SHOW that whenever
we are given an epsilon we are able to produce
a delta that makes the formal definition true.
NOTE: we DON’T know that the inequality with the
epsilon is true – it is the purpose of the proof to
show that it is true!

5. The Given
Informally
• We want x to approach 7.
• We need to show that 10x approaches 70.
Formally
• We’re given an arbitrary epsilon > 0
[that is it; nothing else!]

6. To Show
We need to SHOW that the formal definition
of limit can be satisfied with that arbitrary
epsilon given to us. In formal words,
SHOW there exists a δ>0 so that when x
is chosen so |x-7| < δ, then |10x-70| < ε is
true for all choices of x satisfying that
inequality.

7. The Proof Begins
We must show that |10x-70| < ε.
Note |10x-70| = |10||x – 7| = 10|x – 7|.
So showing |10x-70| < ε is the same as showing
10|x – 7| < ε.
And this is the same as showing |x – 7| < ε/10.
Therefore, once we show |x – 7| < ε/10 is true, that
will be logically equivalent to showing |10x-70| <
ε is true.
Showing |10x-70| < ε is true finishes our proof, so
showing |x – 7| < ε/10 is true would also finish
our proof.

8. The Proof Concludes
Remember we need SHOW there exists a δ>0 so that
when x is chosen so |x-7| < δ, then |10x-70| < ε is true.
So the whole point is to carefully pick δ.
By the work on the previous slide, we will have shown what
we need (|10x-70| < ε being true) when we show |x – 7|
< ε/10 is true.
So the “trick” is to pick δ = ε/10.
So δ exists and is greater than zero,
And when |x-7| < δ is true, we note that δ = ε/10 so
|x – 7| < ε/10 is also true. But that implies |10x-70| < ε is
true, too, and THAT statement is what we were
supposed to show. So the proof is done.

9. A Formal Proof
A formal proof would present the algebraic
arguments, with justifications, of the work
presented on the previous two slides. It
would be more condensed and show the
outcome of all that thinking – it would not
necessarily go through all the explanatory
material. It would still include explanations
of what algebraic steps were done, but
perhaps a lot less on why.

10. Explorations
1. Think about how little the proof would change if the limit
was 80 as x approached 8.
2. Think about how little the proof would change if you were
given a specific value of ε. For example, if you were given
ε=0.1, you can follow the formal steps to show δ=0.01
works.
3. Think about how the algebra of the proof would change (but
not the essence of the proof) if the function was 10x+6 and
the limit was 76 as x approached 7.
4. Note that the algebra becomes MUCH more intense when
the function is nonlinear – far beyond the scope of this
presentation, but certainly within the realm of careful
Google and YouTube searches.