Does a Limit Exist if f(5) Does Not?

[nejnadusho]

If I have a function let's say lim x->5 f(x) that

f(5) Does not exist, because if I give a value of x which equals the limit , there is no anymore limit?


I hope the question is understandble.

Thanks in advance.

 

[quantumdude]

The existence of a limit of a function $f(x)$ at a point $x=c$ has nothing to do with whether the function is actually defined there. That's because a control is built into the definition of a limit that prevents $x$ from taking on the value $c$ as we take the limit. That's what the $0<|x-c|<\delta$ is there for, to ensure that the distance between $x$ and $c$ is strictly positive.

 

[OrbitalPower]

Yeah, I like to think of it as making sure that it is approaching c from both sides.

A good example might be $$\frac{\sqrt{x+1} -1}{x}$$. The limit exists at c > 0 and the answer is 0 but the you can't just plug that in.

if you let x = -0.1 f(x) is 5.132 and -0.01 f(x) .5013, and -0.001 f(x) is .5001 (from the left), and x is .001, f(x) is 0.4999, x is 0.1, f(x) is 0.4988 (from the right), and so on.

When you study continuity I think it really helps clear things up in regards to the existence of limits.

 

[nejnadusho]

Thank you very much