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aiskarl
give an example
give an example to show that \[\lim_{x \rightarrow a} f(x)\] and \[\lim_{x \rightarrow a} g(x)\] doesnot exist but \[\lim_{x \rightarrow a} (f(x)+g(x))\] exist
try this \[ \lim_{x \to 0}{\sin^2x x \over x^2}\] expand sin^2 x as 1 + cos^2x
let f(x)=x and g(x)=-(x+1) and a=infinity then \[\lim_{x \rightarrow \infty } f(x)=\infty \] and \[ \lim_{x \rightarrow \infty } g(x)=-\infty\] but \[ \lim_{x \rightarrow \infty}f(x)+g(x)=-1\]
try something like of this form where you can take LCM and use L'Hopital's rule \[ \infty - \infty \]
@REMAINDER 's example is also good example.
Woops!! \[ \lim_{x \to 0}{\sin^2x \over x^2} = \lim_{x \to 0}{1 \over x^2}+\lim_{x \to 0}{\cos^2 x \over x^2}\]
both 1/x^2 and cos^2x/x^2 does not exist but the sum exists
how to know exist or not
try finding them individually ...
http://www.wolframalpha.com/input/?i=lim+x-%3E0+cos^2x%2Fx^2