Assume that x^4≤f(x)≤x^2 for x include [-1; 1] and x^2≤f(x)≤x^4   for x<-1 and x>1 For which c does this yield a conclusion forlim  f(x)?x→c

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lfryerda eNotes educator| Certified Educator
This question is asking for what value(s) of c does the limit `lim_{x->c}f(x)` exist.  A limit exists if both its right-sided and left-sided limits exist and equal each other.
Now, since we have the inequalities on f(x), we can only assert the limit `lim_{x->c}f(x)` exists when `x ne +-1` provided there are no discontinuities in f(x) in the regions `(-1,1)` , `(-infty,-1)` and `(1,infty)` .  This means that the limit exists for c as any real number in those regions, depending on the values of f(x).
On the other hand, `f(+-1)=1` from the defined inequalities, which guarantees that when `c=+-1` , the limits exist.
The limits are guaranteed to exist when `c=+-1` .