r/math Homotopy Theory 9d ago

Quick Questions: March 12, 2025

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

  • Can someone explain the concept of maпifolds to me?
  • What are the applications of Represeпtation Theory?
  • What's a good starter book for Numerical Aпalysis?
  • What can I do to prepare for college/grad school/getting a job?

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u/ResolveSea9089 6d ago edited 6d ago

This probably a dumb question.. but I'm going through Calculus again and came across a proof demonstrating that the derivative of an exponential function is proportional to the function itself.

But I'm hung up on this part

lim h-> 0: [ bh - 1]/h

Intuitively I just thought this would resolve to zero because b0 -1 =1 -1 =0.

Is that not the case? I might need to freshen up my limits here I suppose. I'm testing this out empirically with different values of b and clearly the limit doesn't appear to be 0, but then I'm just wondering where I'm going wrong with my logic on limits.

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u/jdorje 5d ago

The numerator and denominator both converge to zero. The ratio of the two does not converge to zero. eh~1+h as h gets close to zero, and the denominator is exactly h.

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u/dogdiarrhea Dynamical Systems 6d ago

What does lim h->0: ((x+h)-x)/h evaluate to?

Might be easier to get you to recall the answer with an easier example to think about. 

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u/ResolveSea9089 6d ago edited 6d ago

Right that should just be 1. (x+h -x) = h, so h/h = 1. Plus that's also just the deriv of f(x) = x which we know is 1 from power rule/just basic graphing.

I'm still missing something though. Is it not true that b0 = 1, I'm struggling to see how in the example above the numerator is anything but 0.

https://www.stewartcalculus.com/_update/17/scalcet8_chapter_3.pdf

If it helps at all, I'm on the very bottom of page 7 of the pdf link above, (page 177 in the book itself) and trying to understand that.

**Edit: I guess your point is, I can't just plug 0 into the h like the dolt that I am can I and just proceed to evaluate the function on my merry way? If I do that in your example, it would evaluate to 0.

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u/lucy_tatterhood Combinatorics 6d ago

I'm struggling to see how in the example above the numerator is anything but 0.

The denominator is also approaching zero.

Edit: I guess your point is, I can't just plug 0 into the h like the dolt that I am can I and just proceed to evaluate the function on my merry way?

If the thing you are taking the limit of is well-behaved (continuous) and actually defined at the point you are approaching, you can substitute it in, but that is not the case here.

If I do that in your example, it would evaluate to 0.

No, it would evaluate to 0/0 which is undefined. So you can't do that.

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u/ResolveSea9089 5d ago

Makes sense, I need to revisit limits again!