1. Help with Advanced theoretical Limits #1

I was wondering if someone could help me with the problems within the picture:

I am not looking for straight up answers (unless of course someone is willing to do so), but just for some help finding resources on how to go about solving these problems. Thx in advance for all any and all help!

2. Originally Posted by Some1Godlier
I was wondering if someone could help me with the problems within the picture:

I am not looking for straight up answers (unless of course someone is willing to do so), but just for some help finding resources on how to go about solving these problems. Thx in advance for all any and all help!
1. You are told that $f(x)$ is continuous on all $x$ and that $f(p) = 0$.

What do you think $\lim_{x \to p}f(x)$ is?

2. Hint: $\lim_{x \to a}[f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)$.

3. Hint: By the null factor law, if $A \cdot B = 0$ then $A = 0$ or $B= 0$ or $A = B = 0$.

Hint: $\lim_{x \to a}[f(x)\cdot g(x)] = \lim_{x \to a}f(x)\cdot\lim_{x\to a}g(x)$.

3. 1. 0?
2. +INFINITY
3. 0? Not sure about this because if limit of f(x)= 0 then the limit of f(x) x g(x) would be indeterminant; we can't say that 0 times infinity = 0

4. Originally Posted by Some1Godlier
1. 0?
2. +INFINITY
3. 0? Not sure about this because if limit of f(x)= 0 then the limit of f(x) x g(x) would be indeterminant; we can't say that 0 times infinity = 0
1. Correct

2. Correct

3. Correct - and yes you can. 0 times ANYTHING is still 0, even if the number is infinitely big.

5. Really appreciate it my friend! Now....Could you help me with my other threads....?? If you could I'd be really grateful.

6. Originally Posted by Prove It
[snip]
3. Correct - and yes you can. 0 times ANYTHING is still 0, even if the number is infinitely big.
It has to be zero but not for this reason. $0 \times \infty$ is an indeterminant form. Limits leading to indeterminant forms like this are not necessarily equal to zero.

For #3, the reason is that the only way the limit can be as given is if f(x).g(x) has the above indeterminant form.