Limits of functions at infinity

Juggler123
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Suppose that a function f:R to (0,infinity) has the property that f(x) tends 0 as x tends to infinity. Prove that 1/f(x) tends to infinity as x tends to infinity.

I don't really know where to start with this problem, I'm assuming it will involve some sort of epsilon proof but that's all I know. Please any help would be great!
 
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Write out what it means when you say that f(x) tends to zero as x tends to infinity, and think about what that means for 1/f(x).
 
If f(x) tends to zero as x tends to infinity it means that as x gets progressively larger f(x) gets smaller and smaller thus meaning that 1/f(x) gets larger and lerger but this a proof.
 
No, write out the meaning of f(x) going to zero in "epsilon" form. You have the right outline in your second post, but it isn't a proof ("hand waving" would be the description). Start with "\lim_{x \to \infty} f(x) = 0 means that for any \epsilon > 0 ..." and continue.
 
the function you have given is defined in certain interval suppose take the generalised soln into consideration and prove for soln
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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