Hey! For the last example ( the one you told us to think about it) I was wondering if we suppose that it is actually possible to have cos(teta)+sin(teta)=0 if we multiply by r we gonna find x+y=0 but this can't happen since it's not in the domain of the function itself correct?
That is right! All points (x,y) such that x+y=0 are not part of the domain of this function. As such, the proper (epsilon/delta) definition of the limit would not consider these in the first place.
A bit late to comment on this video, but for the last function we actually don't have a limit. Consider a parametrization of (x, y) = (t, t^n - t), where n is a positive real number; when t goes to 0, so will both x and y. Yet, if you plug these values into (x^2 + y^2)/(x + y), you will find that this evaluates to 2t^(2 - n) + t^n -2t, which depending on the value of n will give a different value for the limit: * 0 < n < 2 gives a limit of 0 * n = 2 gives a limit of 2 * n > 3 creates a division by 0 Which therefore means that we cannot assign a value to that limit.
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Hey! For the last example ( the one you told us to think about it) I was wondering if we suppose that it is actually possible to have cos(teta)+sin(teta)=0 if we multiply by r we gonna find x+y=0 but this can't happen since it's not in the domain of the function itself correct?
That is right! All points (x,y) such that x+y=0 are not part of the domain of this function. As such, the proper (epsilon/delta) definition of the limit would not consider these in the first place.
A bit late to comment on this video, but for the last function we actually don't have a limit.
Consider a parametrization of (x, y) = (t, t^n - t), where n is a positive real number; when t goes to 0, so will both x and y.
Yet, if you plug these values into (x^2 + y^2)/(x + y), you will find that this evaluates to 2t^(2 - n) + t^n -2t, which depending on the value of n will give a different value for the limit:
* 0 < n < 2 gives a limit of 0
* n = 2 gives a limit of 2
* n > 3 creates a division by 0
Which therefore means that we cannot assign a value to that limit.
But I don't understand why r must be from positive side, why not from all side sides?
It’s because r designates the radius in polar coordinates so it can’t be < 0