26: The squeeze should go
For the sake of completeness, the fact that
x/2x
tends to zero should be proved, but that is a textbook example for the
l'Hospital rule. Note that it was just proved that the power
2x2 is even higher on the scale of powers than
xx.
What still remains is to put the "−1" in somehow. Note that comparison does
not help here, since
xx/(2x2 − 1)
is
larger than
xx/2x2. The fact that a
smaller term goes to zero means nothing for the larger one, it can go
anywhere (well, anywhere in the positive area, including infinity and DNE).
This is a nice example of a comparison that does not allow any conclusion.
What to do?
Try the usual trick, factoring out the leading terms. You just worked out how
the leading terms compare, so there should be no trouble.
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