Formal Definition of a Limit
Formal Definition of a Limit
Can anyone here explain to me why/how the formal definition of a limit proves that it exists?
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CDN_Merlin
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Immortal Lobster
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it has a formal definition? lol
basically, a limit exists when there is defined location on a graph that can not be reached by it, or a point that if you were to follow a lone without lifting your pencil, you flatout have to stop due to gap in the path...
Kind of like driving on a road, the limit is where the road ends on a cliff, you cant magically drive over that cliff to get to the otherside without leaving the road
...this all sounded better in my head
basically, a limit exists when there is defined location on a graph that can not be reached by it, or a point that if you were to follow a lone without lifting your pencil, you flatout have to stop due to gap in the path...
Kind of like driving on a road, the limit is where the road ends on a cliff, you cant magically drive over that cliff to get to the otherside without leaving the road
...this all sounded better in my head
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Kilarin
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GOOD one Dedman!
See http://mathforum.org/library/drmath/view/53403.html For an in depth discussion on the formal definition of a limit and what it means and what it tells us.
See http://mathforum.org/library/drmath/view/53403.html For an in depth discussion on the formal definition of a limit and what it means and what it tells us.
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Lothar
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How formal is the definition of a limit you're using? Calculus, or Real Analysis? (That is, are you looking for me to explain epsilon-deltas or completeness?)
The calculus definition of a limit (of a function) says that, if
lim f(x) = b
x->a
then, for any epsilon>0, I can find delta>0 such that if |x-a|<delta then |f(x)-b|<epsilon.
In other words, if you tell me \"you have to get THIS CLOSE to the limit\" I can show that it can be done just by getting close enough to the value. If you tell me \"I want f(x) to be within .00001 of b\" I can say \"sure, as long as x is within .00000000001 of a\" or something similar. If that's the case, then it seems quite straightforward to me that the limit is \"b\". As x gets close to a, f(x) gets close to b. You can think of the argument in geometric terms as a box with height 2*epsilon and width 2*delta, centered around the point (a,b). The \"limit\" simply says that no matter how short you make the box, I can make it narrow enough that the function stays inside the top/bottom boundaries of the box.
A similar argument applies for limits of sequences, limits at infinity, one-sided limits, etc.
But, you might be asking the Real Analysis question -- how do we know there aren't \"holes\" in the real numbers, the same way as there are \"holes\" in the rationals? That is, how do we know that \"b\" (from above) is a real number, rather than something that falls in between real numbers? It turns out the real numbers are defined as \"complete\" -- that is, every Cauchy sequence (every sequence wherein all \"tail\" terms get arbitrairly close to each other) is guaranteed to converge. In other words, the real numbers are simply defined not to have holes in between -- if we keep getting closer and closer to somewhere, that \"somewhere\" must be a real number too.
The calculus definition of a limit (of a function) says that, if
lim f(x) = b
x->a
then, for any epsilon>0, I can find delta>0 such that if |x-a|<delta then |f(x)-b|<epsilon.
In other words, if you tell me \"you have to get THIS CLOSE to the limit\" I can show that it can be done just by getting close enough to the value. If you tell me \"I want f(x) to be within .00001 of b\" I can say \"sure, as long as x is within .00000000001 of a\" or something similar. If that's the case, then it seems quite straightforward to me that the limit is \"b\". As x gets close to a, f(x) gets close to b. You can think of the argument in geometric terms as a box with height 2*epsilon and width 2*delta, centered around the point (a,b). The \"limit\" simply says that no matter how short you make the box, I can make it narrow enough that the function stays inside the top/bottom boundaries of the box.
A similar argument applies for limits of sequences, limits at infinity, one-sided limits, etc.
But, you might be asking the Real Analysis question -- how do we know there aren't \"holes\" in the real numbers, the same way as there are \"holes\" in the rationals? That is, how do we know that \"b\" (from above) is a real number, rather than something that falls in between real numbers? It turns out the real numbers are defined as \"complete\" -- that is, every Cauchy sequence (every sequence wherein all \"tail\" terms get arbitrairly close to each other) is guaranteed to converge. In other words, the real numbers are simply defined not to have holes in between -- if we keep getting closer and closer to somewhere, that \"somewhere\" must be a real number too.
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Foil
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The formal definitions and concepts are the same, but slightly differently applied.
In Calculus, the question is generally, \"Does a limit of function F exist at point x?\" The answer to that question can be a \"No\", as limits don't necessarily exist at every point. Limits are fundamental to Calculus, as everything is built from the derivative of a function, which is defined as a limit.
In Real Analysis, the question is usually more along the lines of, \"If a limit of function F exists, what are the properties of that limit?\" For example, as Lothar explained, the Completeness of the Reals shows that if a limit of a real-number function exists, it will be a real number.
In Calculus, the question is generally, \"Does a limit of function F exist at point x?\" The answer to that question can be a \"No\", as limits don't necessarily exist at every point. Limits are fundamental to Calculus, as everything is built from the derivative of a function, which is defined as a limit.
In Real Analysis, the question is usually more along the lines of, \"If a limit of function F exists, what are the properties of that limit?\" For example, as Lothar explained, the Completeness of the Reals shows that if a limit of a real-number function exists, it will be a real number.
