Limits in Mathematics?

Question

I only ever encountered the limit while learning derivation by first principles in calculus. I understood all the theory behind first principles, but we were never told what happens to the limit h -> 0. Our teacher just said that it goes away after we divide by h, and that’s all I got.

I understand that the limit h -> 0 represents the gap between x and (x + h) getting smaller and smaller. But how does this gap disappear at the end? From searching online I’ve learned that limits are not *equality*, h never *equals* zero, it just gets closer and closer to it. But then why does it equal zero at the end? How is h -> 0 no longer intrinsic to f'(x)? This might be a dumb question but it has stumped me for ages now.

In: 8

Answer 1

There are two ways to answer one is handwavey the other is how it’s done formally in maths.

Handwavy: first you need to convince yourself that a limit from an infinite sequence of steps actually can exist.

Consider the number 2. Well it’s real right? You can count it, there it is. Now consider the sequence 1.9 1.99 1.999 1.9999… etc. This sequence has nth term 1 + sum k=1 to n (9×10^-k)

The infinite sequence has a limit it is 2. As n increases the sequence converges closer and closer to 2. We say the limit as n tends to infinity is 2. Asking what the limit of the gradient chord is as h gets smaller and smaller is similar. The gradient chord has a gradient for each h. It approaches the tangent gradient (as h to 0) in a similar way to the sequence 1.999…n times approaches 2.

The reason why you can effectively make h zero at the end of differentiation first principles is because at the start (with h=0) you have 0/0 which is an indeterminate form. At the end of the process you have the gradient function plus an expression multiple by a power of h. This can be evaluated if h is zero.

For instance x^3 from first principals ends with 3x^2 + 3xh + h^2. If h is zero here the last two terms disappear.

Now the above is handwavey. The proper way is via mathematical real provides rigorous epsilon delta definitions of limits etc. If you follow this logic it is clear why a limit is attained but you do lose some intuition.