Question

I am a bit rusty on this due to summer break, and help would be appreciated. Thank you. What is a limit? Describe the difference between lim f(x) as x approaches c and f(x). How are limits relates to continuity? How are limits calculated?

Answer 1

`What is a limit?`
Intuitively and informally, a limit is where you have x slowly get closer and closer to a certain value. At the same time, you're looking to see how f(x) or y is behaving. If y is getting closer and closer to a fixed value, then we say that the limit exists. If not, then the limit doesn't exist.

These pages may help

https://www.mathsisfun.com/calculus/limits.html
http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx

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There's a more formal definition using epsilon and delta. That requires a lot of explanation really that this page does a better job than what I can do

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html

the basic idea is that you set a tolerance to how close you want to be for epsilon. That in turn will set up delta

Answer 2

`Describe the difference between lim f(x) as x approaches c and f(x).`
I'm not sure I understand what you mean here?

Answer 3

`How are limits relates to continuity?`

if you can show that...

1) The limit exists at x = a
2) The function is defined at x = a (ie f(a) exists)
3) the limit at x = a is equal to f(a)

then that will show the function is continuous at x = a

Answer 4

`How are limits calculated?`

Most are done by simple substitution (replace x with a value). Sometimes you'll have to do special tricks to simplify or rewrite it into a different form. If you're dealing with infinity, then tables often help. It depends on the specific problem.

Answer 5

@jim_thompson5910 Thank you for your help :) I appreciate it.
And what I mean is what is the difference between having lim f(x) with x--> c underneath it and between the function f(x)

Answer 6

That's the notation used to write limits.

If you wanted to say "the limit as x approaches 2 for f(x)" then you would write \[\Large \lim_{x \to 2}f(x)\]

Answer 7

Okay, this answer, as already mentioned, is given on many, many websites. But I'll try to give an abstract explanation, something you can visualize. Imagine that you and your friend are trying to climb a mountain, but from the opposite sides. Also imagine that this particular mountain has a really sharp peak onto which no one or nothing can stand, it is bound to fall off.

So let us call your side of the mountain 'left' and your friend's side 'right'. Both of you try really hard to reach the peak, almost reach it, but cannot stand on it. Every time you get infinitely close, but cannot stand on that peak.

That is how a limit works. When we say ,
limx→af(x)=L

we mean that as you approach the peak either from the left or the right, your, say, height approaches a certain value. Of course, your height is a direct function of how close you reach to the peak.

See, this explanation isn't what limits are, it's just some visualization.

Now, some mathematics. When we talk about limits, we say that as the input approaches a certain value ( but doesn't actually equal that value) the output approaches 'L' .

What's a function?

Well, you could call it an engine, machine, food processor, anything.
You give some input to it ( the 'x' in f(x) ) and it produces some output. Clearly, x is the oil, f is the machine or engine, and what you get ( say f(x) = x+2 ) is the work that machine does, or it's output.

Now, if your machine's graph doesn't break anywhere, or let's say, you can draw it's graph without lifting your hand off the paper in an interval, then the function is continuous in that interval. Not necessary it'll be continuous everywhere.

I guess other sites can explain how to calculate limits better than me, it's a set of rules and some hard work which is required to calculate them.

But let us say, you want to join two pieces of thread. To check they're joined, ( i.e , your function is continuous) you'll just check the joint's right and left ends for any breakage. The same way, we check a function's continuity at a point by taking it's left and right hand limit and seeing if they're equal. Of course, there will be only one peak, and you and your friend will of course encounter the same peak, if it's one mountain we're talking of, whether from the right or left.

Hope I've explained it to your satisfaction.