Question

give an example

Answer 1

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

Answer 2

try this
\[ \lim_{x \to 0}{\sin^2x x \over x^2}\]
expand sin^2 x as 1 + cos^2x

Answer 3

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
\[ \lim_{x \rightarrow \infty}f(x)+g(x)=-1\]

Answer 4

try something like of this form where you can take LCM and use L'Hopital's rule
\[ \infty - \infty \]

Answer 5

@REMAINDER 's example is also good example.

Answer 6

Woops!!
\[ \lim_{x \to 0}{\sin^2x \over x^2} = \lim_{x \to 0}{1 \over x^2}+\lim_{x \to 0}{\cos^2 x \over x^2}\]

Answer 7

???

Answer 8

both
1/x^2 and cos^2x/x^2 does not exist but the sum exists

Answer 9

how to know exist or not

Answer 10

try finding them individually ...

Answer 11

http://www.wolframalpha.com/input/?i=lim+x-%3E0+cos^2x%2Fx^2