Inverses of asymptotic functions

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Suppose f(x) and g(x) are monotone increasing functions (continuous, and smooth if necessary) which are asymptotic -- that is, their quotient has limit 1 as x→∞. Are their inverses asymptotic?
 
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Hint: Look for counter-examples.
 
Right, mfb! For example, log x and log 2x. What I should have asked was, under what circumstances...
 
That is nearly the example I found (log x and 1 + log x). I think you need those slowly/quickly growing functions, where a small difference in one variable (vanishes in the limit) can lead to a large difference in the other one.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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