limit problem

[shanice12]

my question is how to prove that the limit as x-->a [f(x)+g(x)] may exist even though the limit as x-->a of f(x) and the limit x-->a of g(x) do not exist

i cant seem to find an equation for f(x) or g(x) to satisfy the problem :|

 

[BigGlenntheHeavy]

\(\displaystyle If \ f \ and \ g \ are \ functions \ whose \ limits \ exist \ as \ x \ \implies \ a, \ then:\)

\(\displaystyle \lim_{x\to a}(f+g)(x) \ = \ \lim_{x\to a}[f(x)+g(x)] \ = \ \lim_{x\to a}f(x) \ + \ \lim_{x\to a}g(x)\)

\(\displaystyle This \ should \ help.\)

\(\displaystyle f(x) \ = \ -\frac{1}{x} \ and \ g(x) \ = \ \frac{1}{x} \ as \ x \ \implies \ 0.\)

\(\displaystyle This \ should \ tell \ you \ that \ the \ converse \ may \ or \ may \ not \ be \ true.\)

 

[shanice12]

Thank you that helped a lot