the "|e^x - 0| < epsilon" is just copying from the general definition of "limit of f(x) as x approaches negative infinity equals L"; in general the last part of the statement would say "|f(x) - L| < epsilon" so L is just replaced with 0 you could just say "e^x < epsilon" instead of "|e^x - 0| < epsilon" in this case since both statements are equivalent
This isnt a critique or anything, this is the usual proof youll find of this fact (or a version of it); i have a question. The hypotheses of IVT require that our function be from I to R, where I is a subset of R. But the hypotheses of your proven statement give that our function is from R to (0, infty). How can we justify invoking IVT here?
Of course expanding the codomain from (0, infty) to R is fine. To get the domain as an interval I, simply restrict the domain of e^x to [a, b]. This restriction will still be continuous on [a, b], so we may apply IVT to the restriction.
@frankqiang6318 Using R as the domain is fine, the issue is the codomain. The exponential function is not surjective or "onto" when considered as a function from R to R. So my question remains, how can we justify invoking IVT here? Edit: I know this proof works, I was moreso curious how the creator of the video would tackle this question
It's had to understand why you don't have more subscribers.
Most people who "like math" actually like calculus and solving problems. Only a few prefer rigorous proofs. But I'm really glad this guy does it.
In your definitions, why take the absolute value of e^x - 0? What’s the reason for subtracting 0 (love your content btw)
the "|e^x - 0| < epsilon" is just copying from the general definition of "limit of f(x) as x approaches negative infinity equals L"; in general the last part of the statement would say "|f(x) - L| < epsilon" so L is just replaced with 0
you could just say "e^x < epsilon" instead of "|e^x - 0| < epsilon" in this case since both statements are equivalent
Thank you!
Why do you put brackets around the "know" statements?
This isnt a critique or anything, this is the usual proof youll find of this fact (or a version of it); i have a question. The hypotheses of IVT require that our function be from I to R, where I is a subset of R. But the hypotheses of your proven statement give that our function is from R to (0, infty). How can we justify invoking IVT here?
Of course expanding the codomain from (0, infty) to R is fine. To get the domain as an interval I, simply restrict the domain of e^x to [a, b]. This restriction will still be continuous on [a, b], so we may apply IVT to the restriction.
@frankqiang6318 Using R as the domain is fine, the issue is the codomain. The exponential function is not surjective or "onto" when considered as a function from R to R. So my question remains, how can we justify invoking IVT here? Edit: I know this proof works, I was moreso curious how the creator of the video would tackle this question
This is weird notation in your "Theorem". You dont wreite exp(x):R -> R, but f: R -> R, x |-> exp(x)