Question

Can someone explain to me the definition of a limit?

Answer 1

This is a rough sketch of what I'm about to explain

Answer 2

Limits are values that the graph of a function (can or sometimes doesn't) approach. In most cases in your text the values will not be achieved

Anyways limits are fundamental to the topics of calculus in that they are basically trying to get at the fact the graph can, in a sense, get infinitely close to an x-value (in this case 3) and approach a value (in this the y-value of 2). The graph doesn't actually go to 2 at all it never does. There's a hole there (that just means it defined) and the value at 3 is some negative number (represented by the dot)

Basically a limit, expressed in this notation \[\Large \lim_{x \rightarrow a}=L\]
read (the limit as x approaches a is L means that the graph APPROACHES a value of L as you get infinitely close to that number (a) but doesn't have to be at that value (this is imp. as sometimes the functions does equal the limit but that's if the function is defined there)

In this case of the graph \[\Large \lim_{x \rightarrow 3}=2 \]

Remember, the graph doesn't have to EQUAL 2 at x=3, it just has to APPROACH the value from both sides (I'll get into what if it doesn't from sides in a bit)

Answer 3

From both sides*, do you follow @themathninja

Answer 4

im reading...

Answer 5

I know, take your time, most people sometimes leave the question and never see there's a response

Answer 6

lol i dont get it

Answer 7

OK, I'll set the graph down here again

Answer 8

At x=1, there's a value right?
At x=2, there's a value
As you continue getting bigger values of 2 (2.1, 2.3 2.5, 2.9, 2.9999) basically as you get to 3, you're getting close to the hole right ?

Answer 9

so the limit is saying that when x approaches 3, the "limit" is at 2, which is the hole?

Answer 10

Yes. There are many types of limits, but this one is the most basic to understand

The \[\Large \lim_{x \rightarrow 3}=2 \]

Answer 11

The one that I showed you first was the most basic to understand****

This new drawing has the same limit. BUt htis time the function is actually defined at 3, it doesn;t have to be, but like the first graph, as you approach 3 from both sides and both sides means as I go from

(the left side): 2.5, 2.9, 2.91, 2.99, 2.999
& (the right side): 3.5, 3.2, 3.1, 3.001, 3.000001
I get really close (and actually in this case get) to 2

Answer 12

oh i see, so how do you know if there is a hole or not?

Answer 13

Well, you're gonna have to know to think analytically:

1. If you're given a graph without an equa tion, you'll see it
2. If you're given a graph with an equation and the scale is not that big, you might want to go through and find some holes (I'll show you if you don't know)
3. If you just get an equation, you'll have to find the asymptotes, end-behavior, holes, etc. to know how your graph behaves


May I ask what math you're in to first start you off?

Answer 14

I'm starting ap calc bc

Answer 15

Haha, so am I! Are you on flvs?

Answer 16

understanding the precise definition of limit of a function requires some serious and intense self explanation and insight. Since you're in calculus 1, It suffices to understand intuitively. That is limit, if exists, is pretty much the value which the function is approaching at a certain value of the independent variable

lim x^2 = 4 simply means as x approaches 2 from both sides, x^2 will
x->2

approaches 4. This makes sense if you keep plug values closer to 2. For example, 1.9, 1.99, 1.999 or 2.1, 2.01, 2.001, then when you square them, what you'll get are values closer and closer to 4

Of course, from the pure aspect, it's alot more abstract. In fact, many calculus student who are first exposed to the idea of limit struggle a lot when asked to prove the limit of a certain function

Answer 17

what's flvs? @doulikepiecauseidont

Answer 18

Ok, nvm. It's an online course used in Florida. But i guess you don't use it. Some people on here do though.