Some basic conceptual doubts about limits

[ythamsten]

(1)If the limit of f(x), as x approaches a point a doesn't exist, and the limit of g(x), as x approaches a point a doesn't exist either, the limit of (f+g)(x) and (fxg)(x), as x approaches a point a can possibly exist?

(2)If the limit of f(x) as x approaches a and the limit of (f+g)(x), as x approaches a both exists, so the limit of g(x), as x approaches a, has to exist?

Thanks.

 

[jbunniii]

If the limit of f(x), as x approaches a point a doesn't exist, and the limit of g(x), as x approaches a point a doesn't exist either, the limit of (f+g)(x) and (fxg)(x), as x approaches a point a can possibly exist?

Try to come up with an example where the bad behavior of $f$ and $g$ is canceled out when you form the sum $f+g$. Try to come up with another example that does the same for the product $fg$. Can find an example which works for both $f+g$ and $fg$?

If the limit of f(x) as x approaches a and the limit of (f+g)(x), as x approaches a both exists, so the limit of g(x), as x approaches a, has to exist?

$\lim_{x \rightarrow a} f(x)$ and $\lim_{x \rightarrow a}g(x)$ exist, what can you say about $\lim_{x \rightarrow a} (f+g)(x)$?

 

[Dick]

(1)If the limit of f(x), as x approaches a point a doesn't exist, and the limit of g(x), as x approaches a point a doesn't exist either, the limit of (f+g)(x) and (fxg)(x), as x approaches a point a can possibly exist?

(2)If the limit of f(x) as x approaches a and the limit of (f+g)(x), as x approaches a both exists, so the limit of g(x), as x approaches a, has to exist?

Thanks.

Sure. Can you give examples for (1) and maybe sketch a proof for (2)?

 

[ythamsten]

Thank you very much guys, this returned questions were very clarifying and didatic.
I imagine for (1) two piecewise functions, f(x):= -1 if x<0; 1 if x≥0, and g(x):=1 if x< 0; -1 if x≥0. As x approaches 0, they are clearly not limited, but (f+g)(x) has a limit, and it is 0 (as seen by the side limits). And for the product (fg)(x) it works the same way, as x aproaches 0 from the left we have a -1, and from the right, -1, even though the functions don't have a limit.
For question (2), g(x) has to have a limit, because the limit of the sum could be seen as a translation (the limit of the sum is equal to the sum of limits), and if g(x) don't have a limit, but f(x) does, the sum wouldn't have a limit. Is that true? Again, thank you very much.

 

[Dick]

Thank you very much guys, this returned questions were very clarifying and didatic.
I imagine for (1) two piecewise functions, f(x):= -1 if x<0; 1 if x≥0, and g(x):=1 if x< 0; -1 if x≥0. As x approaches 0, they are clearly not limited, but (f+g)(x) has a limit, and it is 0 (as seen by the side limits). And for the product (fg)(x) it works the same way, as x aproaches 0 from the left we have a -1, and from the right, -1, even though the functions don't have a limit.
For question (2), g(x) has to have a limit, because the limit of the sum could be seen as a translation (the limit of the sum is equal to the sum of limits), and if g(x) don't have a limit, but f(x) does, the sum wouldn't have a limit. Is that true? Again, thank you very much.

Close enuf. g=(f+g)-f. If (f+g) and f have limits then their difference does.