Proving Limits of Function f(x) = x^3/abs(x)

[mathgal]

I have the function f(x)=x^3/abs(x)

I think that the following are all true:
lim f(x)= inf.
x->inf

lim f(x)= 0
x-> 0+

lim f(x)=0
x-> 0-

lim f(x)= -inf.
x-> -inf

and

lim f(x)= dne.
x-> 0

I'm not sure about the last one, because I thought that ususally when the limit from the left and the limit from the right are the same, this means that the lim does exist at that number (in this case 0)? But I know this function is not defined at x=0.

Now I need to prove while all these limits are what I have claimed them to be. I'm guessing I need to use the def. of continuity but I'm not sure. Please help!

 

[Mark44]

I have the function f(x)=x^3/abs(x)

I think that the following are all true:
lim f(x)= inf.
x->inf

lim f(x)= 0
x-> 0+

lim f(x)=0
x-> 0-

lim f(x)= -inf.
x-> -inf

and

lim f(x)= dne.
x-> 0

I'm not sure about the last one, because I thought that ususally when the limit from the left and the limit from the right are the same, this means that the lim does exist at that number (in this case 0)? But I know this function is not defined at x=0.

It's not "usually" - it's "always." The two-sided limit of a function exists iff both one-sided limits exist and are the same number. Yes, the function is not defined at x = 0, but that doesn't have any direct bearing on whether the limit exists.

Now I need to prove while all these limits are what I have claimed them to be. I'm guessing I need to use the def. of continuity but I'm not sure. Please help!

You need to use the definition of the limit (with delta and epsilon). You are not proving that the function is continuous - since it's not defined at 0, it's not continuous at 0.