MHB Understanding Limits: Explaining Left and Right Limits in Simple Terms

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The discussion clarifies the concepts of left and right limits in calculus, specifically using the examples of \lim_{x \to 3^+}\frac{1}{x} and \lim_{x \to 3^-}\frac{1}{x}. Approaching from the right (3+) means values like 3.1, while approaching from the left (3-) involves values like 2.9. It is noted that left and right limits can differ, which may occur in cases of jump discontinuities or when one side approaches infinity. An example provided is the Heaviside Function, illustrating how limits can behave differently. Understanding these limits is crucial for analyzing functions and their continuity.
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Could someone explain what things like \displaystyle \lim_{x \to 3^+}\frac{1}{x} and \displaystyle \lim_{x \to 3^-}\frac{1}{x} are supposed to be?

I googled and it gave me epsilon delta stuff. I'm not smart enough for that. (Doh)

I guess I'm asking if anyone could give a dumbed down explanation of sorts. :D
 
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Poly said:
Could someone explain what things like \displaystyle \lim_{x \to 3^+}\frac{1}{x} and \displaystyle \lim_{x \to 3^-}\frac{1}{x} are supposed to be?

I googled and it gave me epsilon delta stuff. I'm not smart enough for that. (Doh)

I guess I'm asking if anyone could give a dumbed down explanation of sorts. :D

Plus means approaching 3 from the right so 3.1, 3.01, 3.001 etc
I bet you can guess what minus means.
 
Why did you start with 3.1?
 
Poly said:
Why did you start with 3.1?

It isn't really a starting point. I was just emphasizing approaching from the right. We could start at 100 but the goal is to approach 3 and get really close to 3. Really close as in 3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
 
Thanks. Did I read that sometimes \lim_{x \to a^{+}}f(x) and \lim_{x \to a^{-}}f(x) can be different? How's that?
 
Poly said:
Thanks. Did I read that sometimes \lim_{x \to a^{+}}f(x) and \lim_{x \to a^{-}}f(x) can be different? How's that?

That is true. You can have jump discontinuities or one side going to neg inf and the other to pos inf.
Look at the Heaviside Function has an example
 
Here is an example you can try:
$$
\lim_{x\to 1}\frac{x^2 - 2x - 3}{x-1}
$$
You need to check the left and right limit.
 

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