Understanding Limits: A Question about Inequalities

[chemistry1]

Hi, I'm having trouble understanding the following fact about limits :
If f(x)<=g(x) for all x on (a,b) (except possibly at c) and a<c<b then,
lim f(x) <= lim g(x)
x -> c x->c
Here's how I interpret the definition : We have two functions f(x) and g(x), and the inequality f(x)<=g(x) hold true for all values that are not c. (That our interval (a,b)) If we were to evaluate the functions at c (considering that we can do it for our two functions.) then the inequality wouldn't hold anymore. (For example, f(x) would be superiro to g(x))
Please tell me if I have any errors.
THank you!
If you want to read more, go here : http://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx

 

[mathman]

The definition includes the phrase "except possibly at c". This means the limit inequalty will hold. At c the inequalty may or may not hold delepnding on the definition.

Example: f(x) = 1 for x ≠ c, f(c) = k. g(x) = 2 for all x. Then the limits as x -> c satisfy f(x) < g(x). However at c it will depend on whether or not k > 2.

 

[chemistry1]

I was wondering, when we consider several functions at once in the same graph, is it ok if this whole is not a function itself ? Do we care about whether this whole is function or not ?

 

[HallsofIvy]

What "whole" are you talking about? How are you combining these "several functions"?

 

[chemistry1]

Nah, its okay, no need for that anymore.