Prove using the formal definition of limits:
lim_x-a g(x) = inf. and,
g(x) ≤f (x) for x-a,
then:
lim_x-a f(x) = inf.

Question

Prove using the formal definition of limits: 
lim_x->a g(x) = inf. and,
g(x) ≤f (x) for x->a, 
then:
lim_x->a f(x) = inf.

anonymous · Answer

Given,$$\lim_{x \rightarrow a} g(x) = \infty $$Then for every $$\epsilon > 0$$there exists $$\delta >0$$ such that$$0< \left| x-a \right| <\delta$$we have$$g(x) > \epsilon$$And we are told that $$\lim_{x \rightarrow a}f(x) \ge \lim_{x \rightarrow a}g(x)$$Thus,$$f(x) \ge g(x) > \epsilon$$whenever,$$0 < \left| x-a \right|<\delta $$Therefore, f(x) also meets the criteria for a limit which goes to infinity. Thus,$$\lim_{x \rightarrow a}f(x)=\infty$$