You are here:

Calculus/Calculus (LIMITS)

Advertisement

Expert: - 11/20/2010


Question
I want the proof of lim x->a+=lim x->a-

Answer
This proof is not always true.

If this proof is true on f(x),
then f(x) is said to be continuous at a.

The proof is best done with epsilon, e, and delta, d.
Given that we are looking at lim(x->x0), if this is true,
it can be said no matter how small |x-x0| < e,
a d can be found such that |f(x) - f(x0)| < d.

The limit would vary depending on the function involved.

I will now give two examples, one of which works and one that doesn't work.

Suppose f(x) = (x^2 - 1)/(x - 1).
The lim(x->1)f(x) = 2.

As x goes down to 1, f(x) goes to 2.
Here we have x, f(x):
0.5, 1.5
0.75, 1.75
0.875, 1.875
0.9375, 1.9375
0.96875, 1.96875
0.984375, 1.984375
0.9921875, 1.9921875
0.99609375, 1.99609375
0.998046875, 1.998046875
  
1.5, 2.5
1.25, 2.25
1.125, 2.125
1.0625, 2.0625
1.03125, 2.03125
1.015625, 2.015625
1.0078125, 2.0078125
1.00390625, 2.00390625
1.001953125, 2.001953125

As can be seen in both the limit where x is less than one and
the limit where x is greater than 1, the function value approaches 2.
However, at x = 2, f(x) is undefined.

As a case where this is not true, take f(x) = x/|x| with x0 = 0.
For x > 0, f(x) = 1; for x < 0, f(x) = -1.

Therefore the lim(x->x0) a+ is not the same as lim(x->x0) a-
when a = 0.

Calculus

All Answers

Answers by Expert:

Ask Experts

Volunteer

Scotto

Expertise

Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

Experience

Experience in the area: I have tutored students in all areas of mathematics for over 25 years. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

Publications
Maybe not a publication, but I have respond to well oveer 7,500 questions on the PC. Well over 2,000 of them have been in calculus.

Education/Credentials
I aquired well over 40 hours of upper division courses. This was well over the number that were required. I graduated with honors in both my BS and MS degree from Oregon State University. I was allowed to jump into a few junior level courses my sophomore year.

Awards and Honors
I have been nominated as the expert of the month several times. All of my scores right now are at least a 9.8 average (out of 10).

Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.

©2012 About.com, a part of The New York Times Company. All rights reserved.