Can adding two undefined limits result in a defined limit?

[azatkgz]

I've answered to test in this way

1)If \lim_{x\rightarrow a}f(x) and \lim_{x\rightarrow a}g(x)
do not exist,then \lim_{x\rightarrow a}(f(x)+g(x)) may exist or not.
2)if \lim_{x\rightarrow a}f(x) and \lim_{x\rightarrow a}(f(x)+g(x)) exists then \lim_{x\rightarrow a}g(x) must exist.

 

[morphism]

And what are your thoughts on the problems?

 

[azatkgz]

1)I think it usually does not exist,but addtion limits of some functions may be any number ,like \frac{|x|}{x}+\frac{|x|}{x}.
2)Here,I thought that if \lim_{x\rightarrow a}g(x) does not exist then
\lim_{x\rightarrow a}(f(x)+g(x)) does not exist also.

 

[JasonRox]

1)I think it usually does not exist,but addtion limits of some functions may be any number ,like \frac{|x|}{x}+\frac{|x|}{x}.
2)Here,I thought that if \lim_{x\rightarrow a}g(x) does not exist then
\lim_{x\rightarrow a}(f(x)+g(x)) does not exist also.

You didn't give a solution the limit in 1). You basically said the limit of a function of x is another function of x, when x approaches something. I find that hard to believe.

You're adding two limits that don't exist. Is it possible that when adding two limits that don't exist to actually exist after adding them? Think in terms of graphs and how the graph looks like when a limit does not exist.

Using the practice from 1), you should be able to handle 2).