Calculus, all content (2017 edition)
Sal finds the limit of a function given its graph. The function isn't defined at the limiting value but that doesn't mean the limit doesn't exist!
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- What does Sal mean at0:47(when he says that a function can be defined while the limit isn't)?(4 votes)
- When we talking about limits we are also regarding the points near it . But, when we are talking about function we mean it's exact value on graph.(1 vote)
- How do we get the parabola? I am able to plot the limit if the graph is given but how do I make the graph with the parabola myself? Please help me out(2 votes)
- It also isn't a parabola, a parabola would be graphed as x squared. This would probably be a sin(x) function. Harley is right, the equation isn't given.(3 votes)
- Can't limits also be defined as some sort of unique summation notation? I've seen something similar regarding integrals.(1 vote)
- Limits cannot be defined using summation notation. They're related when used to discuss integrals, but they're not the same thing.(4 votes)
- The very beginning of the video. I don't understand the graph. How was it graphed like that?(1 vote)
- The graph is not the main point at the beginning. He is just using it to show an example of limits. But to answer your question that graph is from the parent function sin(x) because of its "wave like" feature and because it has (0,0) as a point. If you need any more help understanding concepts please write me back and i will respond as soon as possible. Hope this helps.(3 votes)
- I'm just not understanding the function. How can a function be y=f (x)? What does that mean? Specifically, how can that work as a function?? There are not any numbers!! How can you graph something that doesn't have numbers, only variables?
Do you get the question?(1 vote)
- y = f(x) does not make much sense, we need to either define y, or define x. For example, if we define y = 2, then we have f(x) = 2
This means give me an x... and I'll give you a 2. It's basically a horizontal line at y = 2 on a graph.
Or we can define f(x), such as f(x) = x^2, then we have x^2 = y
Which could mean give me an x, square it, and that's your y.
For your question I think you meant f(x) = x, this is equivalent to y = x IF we define y = f(x) and y represents the y-axis of a graph. Then this means give me an x value, and that x value is the y value as well.
Just imagine f(x) or a function as a simple variable, that's all it really is.
Normally we define f(x) to mean that f(x) is the dependent variable. It's output DEPENDS on the value of x. Anything to the right of the function, or the function definition, represents the independent side, housing the independent variables. in the example:
f(x) = x^2
f(x) is dependent on the value of x, x is the independent variable.
We can also write it like y = x^2
In this case y is the DEPENDENT variable again, and x is the independent variable.
That's really it.
For an independent variable you can just plug in any value you see fit, that's why it's independent, and that is how we graph it. We start shoving in any values we want into independent variables, and then graph the output for the dependent variable.
for the function: f(x) = x + z, for example, you can probably tell that f(x,z) can be any value under the sun! If we let x = 1 and z = 1, then f(x) = 2.
In this case we would need to define z beforehand, but what you get is essentially every line y = x, shifted to the left or right by the value of z. It represents infinitely many diagonal lines in the x,y plane. There's more like this in linear algebra related to independent variables, but for now this is enough!(2 votes)
- How can the function be defined, while the limit is not defined?(1 vote)
- Suppose you have y=tan(x), and add that wherever this function is undefined, (at odd multiples of π/2), it just equals 0. Then the limit as x goes to π/2 does not exist, since the function goes to infinity at π/2. But our function is defined at π/2: we said that it equals 0.(2 votes)
- what does it mean when you say 'The limit does not exist'? What is its significance?(1 vote)
- A limit is a real number that satisfies a certain definition, given a particular function and x-value. If there is no real number that fits the definition, or multiple numbers have equal claim, we say the limit does not exist.(2 votes)
- Does a function include (or not include) x=c when c is undefined in the function? (in this case, it would be like x=-4 is undefined, does that still mean x=-4 is included in this function?)(1 vote)
- Can The limit of a function at some point (X) be infinity ? Isn't this particularly true for reciprocal functions ?(1 vote)
- This is a case where it is very important to be precise with your wording. Suppose you have a function defined as f(x)=1/x for x>0. As x approaches 0, f(x) approaches infinity. Strictly speaking, this is a case where the limit is considered to not exist, since there is no number* that the function tends towards. Some teachers will let you get away with writing lim f(0)=infinity, but others will require you to say that because the function tends towards infinity, the limit does not exist.
*Infinity is not a number, but is instead a sort of amorphus value. As such, the usual techniques and intiutions you have for numbers won't apply.(1 vote)
- [Voiceover] So we have the graph of y equal f of x right over here, and what we want to do is figure out the limit of f of x as x approaches negative four. So what does that mean? Well, a limit is saying, "What does my function approach "as the input of that function approaches, "in this case, negative four? "As the input approaches a value?" And as we see in this example, the function doesn't necessarily have to be even defined at that value. We can see f of negative four, you go to x equals negative four, and you see that f of negative four is undefined. So this is not defined, but as we'll see even though the function isn't defined there, the limit might be defined there. And actually it can go the other way around. Sometimes a function is defined there, but the limit is not, and we'll see that in future videos. But let's just get an understanding here. What's going on as x approaches negative four? As x approaches negative four from values greater than negative four and from values less than negative four. Well let's first think about values greater than negative four. So when x is negative one, this is f of negative one. This is f of negative two, this is f of negative three. This is f of negative 3.5, this is f of negative 3.9, this is f of negative 3.99, this is f of negative 3.99999. And so you can see the value of our function, as x gets closer and closer to negative four from values greater than negative four, seems to be approaching six. And let's see if that's true from the other direction, so from values less than negative four. So this f of negative seven, f of negative six. This is f of negative five, puts us around seven. This is f of negative 4.5, f of negative 4.1, f of negative 4.01, f of negative 4.00001, it looks like it's getting awfully, it's going to be a little bit more than six, so it seems as we get closer and closer to x equal negative four, the value of our function is approaching positive six.