Intro to Limits
The first important calculus idea we encounter is the curious idea of a limit. Now the way we write this, we have the, the abbreviation lim for limit. And we read this expression, the limit of the function f of x as x approaches c. And what this means really is just where it looks like f of x is going when it's in the neighborhood of x equals c. Now, we think about all of our elementary functions like polynomials.
The idea of a limit is not that interesting, because where it looks like a function is going is exactly the same as where it winds up being at f of c. So, all of the elementary functions, nice, smooth, continuous functions, they always go where they look like they're going. And so limits are not all that interesting when we talk about those functions. They become much more interesting when we talk about a function, say like this, that has a hole in it.
Well then it looks like it's going a certain place, even though f of c itself is undefined. We can still say that the limit exists. Also, we could have a function that is a whole, and is defined somewhere else. Let's say, for example, that the coordinates of that whole are 4,2, and the coordinates of this point are 4,3.
Well, then we might say that the limit of this function as x approaches 4 is 2. It looks like its headed to y equals 2. However it turns out, that when we actually plug in 4, f of 4 equals 3. Its actual output at 4 is something different than what it looks like it's approaching. So that is the basic idea of a limit.
Read full transcriptThe idea of a limit is not that interesting, because where it looks like a function is going is exactly the same as where it winds up being at f of c. So, all of the elementary functions, nice, smooth, continuous functions, they always go where they look like they're going. And so limits are not all that interesting when we talk about those functions. They become much more interesting when we talk about a function, say like this, that has a hole in it.
Well then it looks like it's going a certain place, even though f of c itself is undefined. We can still say that the limit exists. Also, we could have a function that is a whole, and is defined somewhere else. Let's say, for example, that the coordinates of that whole are 4,2, and the coordinates of this point are 4,3.
Well, then we might say that the limit of this function as x approaches 4 is 2. It looks like its headed to y equals 2. However it turns out, that when we actually plug in 4, f of 4 equals 3. Its actual output at 4 is something different than what it looks like it's approaching. So that is the basic idea of a limit.