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TooManyHours

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Hi, I have a question about the epsilon / delta definition of limits, for example the limit of x as it approaches c for f(c) = L.

As I understand it, epsilon is basically the number of units on either side of L on the y-axis that makes a range between L + epsilon and L – epsilon with L being in the exact middle of the range.

There is supposed to be a corresponding range on the x-axis around c using delta units with the range between c + delta and c – delta with c being in the exact middle of the range.

And according to the definition, as long as you pick some x value that falls between c + delta and c – delta, the f(x) function will always produce a value that falls between L + epsilon and L – epsilon range. Sounds straight forward enough.

However, doesn’t this imply that if f(c) = L, then f(c + delta) should be L + epsilon and f(c - delta) should be L – epsilon. That's what all the diagrams in the textbooks show. They show graphs with symmetric ranges.

However, a simple example shows that this is not true.

Take the limit of x^2 as x approaches 5 for example. It’s Limit value is 25.

If I choose a delta of 1 unit, my x-axis range goes from 16 [(5-1)^2] to 36 [(5+1)^2).

If I take the absolute value of f(x) – L for x = 4 then 16 – 25 makes an epsilon of 9 units from L.

If I take the absolute value of f(x) – L for x = 6 then 36 – 25 makes an epsilon of 11 units from L.

Clearly the limit is not in the exact middle of this range or epsilon can have two different values.

Does this mean that this epsilon delta definition only works for delta values that are very very close to c, for example you have to pick a delta of .000001 or even smaller to get values that work out a symmetrical range for delta and epsilon?

Thanks

As I understand it, epsilon is basically the number of units on either side of L on the y-axis that makes a range between L + epsilon and L – epsilon with L being in the exact middle of the range.

There is supposed to be a corresponding range on the x-axis around c using delta units with the range between c + delta and c – delta with c being in the exact middle of the range.

And according to the definition, as long as you pick some x value that falls between c + delta and c – delta, the f(x) function will always produce a value that falls between L + epsilon and L – epsilon range. Sounds straight forward enough.

However, doesn’t this imply that if f(c) = L, then f(c + delta) should be L + epsilon and f(c - delta) should be L – epsilon. That's what all the diagrams in the textbooks show. They show graphs with symmetric ranges.

However, a simple example shows that this is not true.

Take the limit of x^2 as x approaches 5 for example. It’s Limit value is 25.

If I choose a delta of 1 unit, my x-axis range goes from 16 [(5-1)^2] to 36 [(5+1)^2).

If I take the absolute value of f(x) – L for x = 4 then 16 – 25 makes an epsilon of 9 units from L.

If I take the absolute value of f(x) – L for x = 6 then 36 – 25 makes an epsilon of 11 units from L.

Clearly the limit is not in the exact middle of this range or epsilon can have two different values.

Does this mean that this epsilon delta definition only works for delta values that are very very close to c, for example you have to pick a delta of .000001 or even smaller to get values that work out a symmetrical range for delta and epsilon?

Thanks

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