Math Help - What is a Limit

1. What is a Limit

I have read the section of my text a couple times and still don't understand, how can I tell whether or not a function has a limit just by looking at it's associated graph

2. Basically, a limit is what it sounds like. A function f(x) has a limit, say b, at x = a if f(x) become closer and closer to b as x becomes closer to a.

The value of f(x) at x = a is irrelevant to whether f has a limit at a. However, if f does have a limit b, then putting f(a) = b makes f continuous at a. So, if f cannot be made continuous at a, it does not have a limit.

Another way to visualize this is to say that you can zoom indefinitely at x = a and still fit the graph of f in some neighborhood of x = a inside your screen. If, at some point, the graph goes through the top or bottom of the screen regardless of how small you make the segment around x = a, the function does not have a limit at a.

when we say that x is close to a we understand that $x\ne a$.

So if $\displaystyle\lim _{x \to a} f(x) = b$ then we know for any $x \approx a\;\& \,x \ne a$ we must have $f(x)\approx b$.

4. L is limit of function f when x approach a iff $\forall \epsilon >0, \exists \delta>0 \, such \,that\, |x-a|<\delta\Rightarrow |f(x)-L|<\epsilon.$

if x in delta neighbourhood of a then f(x) in epsilon neighbourhood of L

5. Originally Posted by MathoMan
L is limit of function f when x approach a iff $\forall \epsilon >0, \exists \delta>0 \, such \,that\, |x-a|<\delta\Rightarrow |f(x)-L|<\epsilon.$
if x in delta neighbourhood of a then f(x) in epsilon neighbourhood of L
@MathoMan
You have miss a crucial bit in the definition.

$\forall \epsilon >0, \exists \delta>0 \text{ such that }0< |x- a|<\delta\Rightarrow |f(x)-L|<\epsilon.$