Question

Is it true that if f(x) < g(x) for all x ∈ (c-ε , c+ε ) and that lim x -> c f(x) and lim x-> c g(x) both exists that lim x->c f(x)< lim x->c g(x)?

Answer 1

No, it is not true. The inequality in the conclusion need not be strict.

Explanation:

Take `f(x) = -x^2` and `g(x)=x^2` and `c=0`.

In this case, we have `lim_(xrarr0)f(x) = lim_(xrarr0)g(x)`

What is true is that under the conditions given, we always have

`lim_(xrarrc)f(x) <= lim_(xrarrc)g(x)`.