Can Limits Avoid Paradoxes in Derivatives and Integrals?

[Red_CCF]

Hi

I know that the common explanation of derivatives is drawing a secant line through a graph and move one point closer to the other where the space between them is infinitesimal. Similarly, area under a graph can be found by finding the areas of individual rectangles with infinitesimally small width and adding the rectangles' areas together. But I've been reading some material that was assigned by my professor that explains the paradoxes in these common explanations. Ex. as we move two points closer and closer together eventually we would get 0/0 as the slope. Can anyone come up with an explanation that avoids such paradoxes?

 

[lurflurf]

The traditional resolution of this problem is the limit. In the definition of a limit all evaluations that are undefined are avoided.
examaple
lim_{n->infinity} 3^-n
we never say 3^-infinity=0
we say for 3^-n can be made as close as desired to 3 by taking n sufficiently large

so
[f(x+h)-f(x)]/h can be made close to f'(x) by selecting h small.

 

[Red_CCF]

The traditional resolution of this problem is the limit. In the definition of a limit all evaluations that are undefined are avoided.
examaple
lim_{n->infinity} 3^-n
we never say 3^-infinity=0
we say for 3^-n can be made as close as desired to 3 by taking n sufficiently large

so
[f(x+h)-f(x)]/h can be made close to f'(x) by selecting h small.

Oh so with the limit, we simply find the slope of a secant line where the two points are infinitismally close together and as a result we get two non zero numbers for x and y for slope calculations and the slope would not be entirely accurate?

 

[slider142]

Oh so with the limit, we simply find the slope of a secant line where the two points are infinitismally close together and as a result we get two non zero numbers for x and y for slope calculations and the slope would not be entirely accurate?

No. The number that we get out of a limit is the least upper bound (supremum) or greatest lower bound (infimum) of a certain set. The guarantee that such numbers exist is built into the definition of the real numbers. If the number is not a rational number, we can easily construct finite decimal approximations using series. Ie., the decimal digits for pi, which was approached by Archimedes as a supremum of the set of areas of inscribed polygons and infimum of the areas of circumscribed polygons.