Advanced Calculus Analysis Question on HW

[Luna86]

Question: a) Suppose that f(x)>=M for all x=/a, and L1=lim x-->a+ f(x) and L2=lim x-->a- f(x) exist. Prove that L1>=M and L2>=M. b) Suppose that f(x)>M for all x=/a, and L1=lim x-->a+ f(x) and L2=lim x-->a- f(x) exist. Must it be true that L1>M and L2>M? Either prove it or provide a counterexample.

 

[Deleted member 4993]

Question: a) Suppose that f(x)>=M for all x=/a, and L1=lim x-->a+ f(x) and L2=lim x-->a- f(x) exist. Prove that L1>=M and L2>=M. b) Suppose that f(x)>M for all x=/a, and L1=lim x-->a+ f(x) and L2=lim x-->a- f(x) exist. Must it be true that L1>M and L2>M? Either prove it or provide a counterexample.

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[pka]

Question: a) Suppose that f(x)>=M for all x=/a, and L1=lim x-->a+ f(x) and L2=lim x-->a- f(x) exist. Prove that L1>=M and L2>=M. b) Suppose that f(x)>M for all x=/a, and L1=lim x-->a+ f(x) and L2=lim x-->a- f(x) exist. Must it be true that L1>M and L2>M? Either prove it or provide a counterexample.

a) Suppose that \(f(x)\ge M,~\forall x\ne a~\&~\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = {L_1}\).
By the definition of right-hand limit, if \(\varepsilon>0(\exists \delta>0)(x \in (a,a+\delta))\left[ {\left| {f(x) - {L_1}} \right| < \varepsilon } \right]\)
Now suppose that \(L_1<M\) then \(\varepsilon=M-L_1>0\) and go for a contradiction.