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## Calculus 1

### Unit 1: Lesson 4

Formal definition of limits (epsilon-delta)

# Formal definition of limits Part 2: building the idea

Some background intuition to make the formal definition of a limit make intuitive sense. Created by Sal Khan.

## Want to join the conversation?

• What does y=f(x) mean? I thought they were the same thing. Sorry if this is a stupid question.
• John, there are no stupid questions. And actually, your question is a very good one.

In a strict mathematic sense, y is just a variable. When someone writes "y=f(x)", it means that the value of y depends on the value of x, which is another variable. That is, for different values of x, there is a function, called f(x), which determines the value of y.

The x variable is therefore called the "independent" variable, while the y variable is called the "dependent" variable because it's value "depends" on the value of x.

So, in short. The expression y=f(x) is basically a formal way of stating which one is the independent variable (in this case, x) and which one is the dependent one.
• why is it that the range of the L is 0.5 and the range around c was 0.25, my point is that will the range for the x-axis always be less than that for the range for the y-axis?
• I'm pretty sure that it depends on the type of F(x) (the function) that you encounter. You can try with basic function at home, draw one and then find a casual range in Y and watch how it work on the X axis; you'll see that depends on the trend of the function.
• At , how did he find that the corresponding range for c is more or less 0.25 when the range for L is more or less 0.5?
• He never defined the function; it's a hypothetical situation, and he randomly chose the interval (aka he could have chosen c + or - any constant here).
• Why were epsilon delta forms created?
• Initially, calculus was based on somewhat vague ideas about infinitesimally small quantities, and some scholars argued that these methods weren't valid. The epsilon-delta concept was created to provide a rigorous logical foundation for the methods used in calculus.
• Are limits used only in situations when x approaches the value for which the function f(x) is undefined, and not for other values?
• Usually, but not necessarily.

It's completely fine to say lim{x->1}x=1, this is proper and completely allowed, but somewhat pointless.
• why should one be able to find range of c to prove that limit exist..?... what is basic idea .?
• it is a set up for the proof in the next video, and gets you thinking.
• In what kind of circumstances would people use this Epsilon-Delta definition out of the studies of mathematics?
• in determining what delta you should choose to make the certain error lim x>c f(c). example: how much delta from c-delta<x<c+delta should i choose to make my square area error 0.5.? (If we want f(x) to get this close to f(c), how close does x have get to c?)
• Around you start talking about the range. What if the graph dramatically spikes within that range? Your graph is between L-0.5 and L+0.5 at all times but what if at L-0.25 it all of the sudden dramatically spikes out of L-0.5 just for a quick dip and comes back? Then it isn't within that range, then you can find an X within that range that doesn't give you a y within that range?

Sorry for the poorly phrased question.. I hope someone understood what I was trying to convey.
• Interesting question. The way I was able to explain it to myself to understand was to change the terminologies all together. I used the concept of domain and ranges rather than using a bounded range around L and C. If you visualize the concept of the range of the function given the domain 0.25 < c < -0.25, you would be able to see that no matter how many spikes occur in that DOMAIN range, the value at that spike will INDEED always map to (or be mapped from?) a value within the original domain which was the boundary around c. You have to, in a way, step back from a calculus way of reasoning a little bit to a more basic algebraic way of reasoning. As long as the value y = f(x) exists (in this case, the value of the spike) within a domain (in this case + or - 0.25 off c), any value within the domain ( boundary around c) will always map to a value within its EQUIVALENT range (which in this case is the boundary around L). Since your spike has been introduced to the function, the range of the function can no longer be + or - 0.5 off L (even though the domain remained the same). The range has to be increased for the spike to be accounted for or else, the guy who is playing the game is trying to cheat by bending the rules of algebra a little bit. :) . Think of it this way, by introducing the spike, that value will cease to exist because it indeed doesn't fall within the range + or - 0.5 around L, therefore we will have to go back to the drawing board and redefine a value that would fall within the range specified.That's the way I was able to understand it. I hope it helped you because it actually made me understand algebra a little bit better which i thought was pretty neat. Math works that way! :)
• If it doesn't lie within the "range", does that mean the limit doesn't exist? Is it like saying if the limit doesn't approach the same point it doesn't exist? Thanks. You guys are awesome.
• yes. If there is no range for c where it gets closer to L within that range, the limit does not exist.