Neat-looking graph as the one I just drew. Limit as x- that's an x- as x approaches 2 of f of x? That's an x. What is the limit- I used cursive this time- what is the Let me to define- let me sayį of x is equal to x squared when, if x does not equal 2,Īnd let's say it equals 3 when x equals 2. Wrinkle on that, and hopefully now you'll start to see what Is equal to 4, and that would have been a no-brainer. Is f of x, that if f of x is equal to x squared, that f of 2 Just stuck 2 in there, and I know that if this is- say this This seems like a useless concept because I could have Value, what does the expression equal? In this case, it equals 4. You move on the curve closer and closer to the expression's This expression equal? Well, it essentiallyĮquals 4, right? The expression is equal to 4. To 2 from this direction, and as we get to x closerĪnd closer to 2 to this direction, what does Pretty obvious, but as x- as we get to x closer and closer To the trouble of learning this new concept because it seems See where this is going and be wondering why we're even going The expression approach? And you might, I think, already Left than 2 and from numbers right than 2, what does Squared is equal to 4, right? So a limit is saying, as xĪpproaches 2, as x approaches 2 from both sides, from numbers Or the expression- because we don't say what Like this, right? And when x is equal to 2, y, If we look at- let meĪt least draw a graph. X approaches 2? Well, this is pretty easy. Value does the expression x squared approach as OK, the limit as xĪpproaches 2 of x squared. Say the limit- oh, my color is on the wrong- OK, let me OK, let's say I had the limit,Īnd I'll explain what a limit is in a second. Make sure I have the right color and my pen works. Well, first an explanation before I do any problems. If you'd like a more solid explanation about the math involved to the problem of the circle and the polygon feel free to tell me and i'll work out the math. Hope this gives you an insight to what you can do with limits and really encourage you to keep learning about this topic in particular. So you could state that as "S" (The number of sides of the polygon) approches infinity, then the area of the polygon approches or is basically the same as the area of the circle. Now let's say "S" denotes the number of sides of the polygon, then you can define a function in wich to determine quite precisely the area of the polygon in terms of "S". However as you add more and more sides to the polygon (imagine an hexagon inside of the circle) then the area of the polygon APPROCHES the area of the circle more and more. If you take a polygon, let's say a square, you can put it inside the circle and the area of the square is going to be somehow close to the area of the circle, but still very far off from the real value. You might say "G, it's imposible!" but actually limits can help with this problem. and you want to calculate the area of a circle. For example, imagine that suddenly the formula to get the area of a circle has been removed from all knowledge, texbooks, etc. If you remember from conic sections we actually use limits to find the assympthotes of the hyperbola, also you can use limits to define other cool stuff. Some specifc examples might also be right here in mathematics.
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