Question

4. Use the ǫ, δ-definition of limits to show that
lim
x!a
f(x) = L if and only if lim
x!a−
f(x) = lim
x!a+
f(x) = L.

Answer 1

Please help me. See screenshot

Answer 2

assume\[\lim_{x\rightarrow a}f(x)=L\]then by definition \[\forall\epsilon>0,\space \exists\space \delta >0,st\\a-\delta<x<a+\delta\implies|f(x)-L|<\epsilon\]its pretty easy to see that the definitions for left/right handed limits are satisfied here.

Answer 3

so the converse is pretty much the same thing, just write out the definitions and the result follows.

Answer 4

Thanks! Could u help me look at this question? It makes me stunned.

Answer 5

Give me a bit, I need cook something fast, then I will come back. Maybe close this and open another just in case others can help while im afk

Answer 6

im not sure this is right...

Answer 7

\[by\space definition\\\forall\epsilon>0,\exists\space \delta_1>0\space s.t.\space|x-a|<\delta_1\space \implies|g(x)-c|<\epsilon\\and\\\forall\space M\in R,\exists\space\delta_0>0\space s.t.\space0<|x-a|<\delta_0\space \implies f(x)>M\\let\space \delta \space=min\{\delta_0,\delta_0\}\\\text{then we have}\\|x-a|<\delta \implies|g(x)-c|<\epsilon \text{ and }f(x)>M\\so\\c-\epsilon<g(x)<c+\epsilon\\M<f(x) \\so\\M+c-\epsilon<f(x)+g(x)\]