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- TL;DR Summary
- Nature of a limit.

Hello. I bought "Calculus Made Easy" by Thompson and it got me thinking about something I wondered about before.

This question is a bit hard for me to articulate, but I'll do my best: When we are trying to find the limit as change in x approaches zero of dy/dx, we take smaller and smaller changes in x and see what the answer is getting closer and closer to. E.g. we can plug in .1 as the change in x, then .01, then .001, etc, and see that are results are getting closer and closer to a certain number.

In the case of f(x) = X^2, we see that the result is getting closer and closer to 2x when we try and work out the derivative by taking smaller and smaller changes in x. How can we ever be sure, though, that it will ever reach 2x? I we taking it on belief/philosophy, or is there a proof? Although change approaches zero, we can't ever really have a change of zero.

This question is a bit hard for me to articulate, but I'll do my best: When we are trying to find the limit as change in x approaches zero of dy/dx, we take smaller and smaller changes in x and see what the answer is getting closer and closer to. E.g. we can plug in .1 as the change in x, then .01, then .001, etc, and see that are results are getting closer and closer to a certain number.

In the case of f(x) = X^2, we see that the result is getting closer and closer to 2x when we try and work out the derivative by taking smaller and smaller changes in x. How can we ever be sure, though, that it will ever reach 2x? I we taking it on belief/philosophy, or is there a proof? Although change approaches zero, we can't ever really have a change of zero.