Let aR and let f and g be real functions defined at all points x in some open in
ID: 1944059 • Letter: L
Question
Let aR and let f and g be real functions defined at all points x in some open interval containing a except possibly at x=a. Decide which of the following are true and which are false. Prove the true ones and give counter examples for the false ones.
A) For each nN, the function (x-a)nsin(f(x)(x-a)-n) has a limit as x->a
B) Suppose that {xn} is a sequence converging to a with xn not equal to a. If f(xn)->L as n->, then f(x)->L as x->a
C) If f and g are finite valued on the open interval (a-1, a+1) and f(x)->0 as x->a, then f(x)g(x)->0 as x->a
D) If limx->a f(x) does not exsist and f(x) less than or equal to g(x) for all x in some open interval I conataining a, then limx->a g(x) doesn't exsist either.
Explanation / Answer
true C) If f and g are finite valued on the open interval (a-1, a+1) and f(x)->0 as x->a, then f(x)g(x)->0 as x->a
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