The Squeezing Theorem (two problems)

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How would you go about with the following problems? What are the first steps? Share on other sites

(Forgive any misuse of terminology here - I haven't formally taken this option yet, and all knowledge I have is from my own extended studies)

I think what it's asking you to do here is find two series g(x) and h(x) - or, in this case, to use two sequences g(x) and h(x) which are definitions for f(x) where n>0 or n<1 respectively - prove that the limit of the nth term of both of those sequences has a limit as n approaches infinity at zero, show that the series are convergent, and show that for all x, g(x)<f(x)<h(x), and finally, since you have satisfied all the criteria to use the theorem, invoke the theorem to make the proof.

For 2(a), it seems like it could be done using a similar method, just taking into account that the values of a*n and b are likely to be insignificant as n approaches infinity. However, it seems like a more difficult problem that would require careful manipulation to produce squeezing functions that use a and b and act as bounds for all a and b within their respective domains. 2(b) should follow from the case a=1, b=0.

I've probably made a mistake somewhere in there, but I hope that gives you an idea where to start.

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