Rate of lim exp(x) ~? rate of lim exp(-x)

[BCox]

Hello:
The actual functional analysis for my rate of convergence is a bit more complicated. But essentially the problem I have is knowing if the following is true:

lim exp(x) -> infinity as x->infinity
is at the same rate as
lim exp(-x) -> 0 as x-> infinity

? Would really appreciate the actual answer. Thank you!

 

[ImAnEngineer]

Rate as in slope of a function?

if
f(x) = e^x
g(x) = e^-x

then
f'(x) = e^x
g'(x) = -e^-x

so
f'(x) / g'(x) = -1

So in absolute value, the rate is the same. Although one is positive and the other negative. Does this answer your question?

 

[BCox]

Yes, it does.

What if my f(x) = e^x + cot y ... where my y is another variable for which cot y may diverge to positive infinity? I am still only interested in the rate for when x -> infinity.

What would g(x) look like for the rate at which the new f(x) approaches infinity at the same rate that g(x) approaches zero?

 

[Preno]

f'(x) = e^x
g'(x) = -e^-x

so
f'(x) / g'(x) = -1

Um, no.

What would g(x) look like for the rate at which the new f(x) approaches infinity at the same rate that g(x) approaches zero?

What is "the rate at which <something> approaches infinity"? I mean, I know what it is for one function to approach infinity faster than another, but what is it for one function to approach infinity faster than another function approaches zero?

 

[BCox]

Um, no.
What is "the rate at which <something> approaches infinity"? I mean, I know what it is for one function to approach infinity faster than another, but what is it for one function to approach infinity faster than another function approaches zero?

Yes, I looked at that more closely and realized that is wrong. The question behind the question is, what is the rate of the following at which the function below approaches zero:


lim (1/y) arccot [ -exp(x)/sin(y) - cot(y) ] ->0 as x->infinity

where y is (-pi,0)

 

[ImAnEngineer]

:yuck: what a nonsense i wrote, sorry