1. ## Evaluate the limits

I have such a problem:

Given that lim x->a f(x) = 0, lim x->a g(x) = 0, lim x->a h(x) = 1, lim x->a p(x) = INF, lim x->a q(x) = INF,

Evaluate these limits:

1. Lim x->a f(x)/g(x) =
2. Lim x->a f(x)/p(x) =
3. Lim x->a h(x)/p(x) =
4. Lim x->a p(x)/f(x) =
5. Lim x->a p(x)/q(x) =

I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we don’t have enough info to determine the limit.

1. Lim x->a f(x)/g(x) = 0/0, (DNE)
2. Lim x->a f(x)/p(x) = 0/INF, (Indeterminate form)
3. Lim x->a h(x)/p(x) = 1/INF, (positive infinitive)
4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
5. Lim x->a p(x)/q(x) = INF/INF, (DNE)

I'm not quite sure in my reasoning so I need your input, thanks

2. ## Re: Evaluate the limits

Originally Posted by dokrbb
I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we don’t have enough info to determine the limit.
Could you remind what an indeterminate form is?

Originally Posted by dokrbb
Lim x->a f(x)/g(x) = 0/0, (DNE)
And what if f(x) = g(x)?

Originally Posted by dokrbb
Lim x->a f(x)/p(x) = 0/INF, (Indeterminate form)
And what if a = 0, f(x) = x and p(x) = 1/x?

Originally Posted by dokrbb
Lim x->a h(x)/p(x) = 1/INF, (positive infinitive)
The bigger the denominator, the bigger the ratio?

Originally Posted by dokrbb
Lim x->a p(x)/q(x) = INF/INF, (DNE)
And what if p(x) = q(x)?

3. ## Re: Evaluate the limits

Originally Posted by emakarov
Could you remind what an indeterminate form is?
it is a result that doesn't provide enough info about the actual limit after replacing the values in the expression,

so, wait a min, if by indeterminate forms is meant 0, 0/0, inf, inf/inf, it means I have some more indeterminate forms in my problem?

Originally Posted by emakarov

And what if f(x) = g(x)
you mean what if 0 = 0, ? but this is also an indeterminate form, isn't it?

Originally Posted by emakarov

And what if a = 0, f(x) = x and p(x) = 1/x?
then f(x)=0 and p(x)=1/0 and are indeterminate forms

Originally Posted by emakarov

The bigger the denominator, the bigger the ratio?
well, actually, the bigger the denominator, more the values of f(x) approaches 0 (from how I understood, lesser and lesser parts of 1 we obtain, e.g. 0,5; 0,001; 0,00001 and so on)

Originally Posted by emakarov
And what if p(x) = q(x)?
but this means that INF = INF
=> indeterminate forms, too?

4. ## Re: Evaluate the limits

Originally Posted by dokrbb
Given that lim x->a f(x) = 0, lim x->a g(x) = 0, lim x->a h(x) = 1, lim x->a p(x) = INF, lim x->a q(x) = INF,
Evaluate these limits:
1. Lim x->a f(x)/g(x) =
2. Lim x->a f(x)/p(x) =
3. Lim x->a h(x)/p(x) =
4. Lim x->a p(x)/f(x) =
5. Lim x->a p(x)/q(x) =

I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we don’t have enough info to determine the limit.
1. Lim x->a f(x)/g(x) = 0/0, (DNE)
2. Lim x->a f(x)/p(x) = 0/INF, (Indeterminate form)
3. Lim x->a h(x)/p(x) = 1/INF, (positive infinitive)
4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
5. Lim x->a p(x)/q(x) = INF/INF, (DNE)
Look I know that I am not as nice an instructor as emakarov.
But I must tell you that I think that you have all of these concepts confused.

Lets look at #2,
You have a fraction. Its numerator is very 'close' to 0.
Its denominator is very large in a positive sense.

Here is an example: $~\frac{10^{-100}}{10^{100}}$.
Of what do you think that fraction is approximation ?

Now you have correctly listed Indeterminate forms, of which #2 is none of.

So can you repost this so that it makes sense?

5. ## Re: Evaluate the limits

Originally Posted by Plato
Look I know that I am not as nice an instructor as emakarov.
But I must tell you that I think that you have all of these concepts confused.

Lets look at #2,
You have a fraction. Its numerator is very 'close' to 0.
Its denominator is very large in a positive sense.

Here is an example: $~\frac{10^{-100}}{10^{100}}$.
Of what do you think that fraction is approximation ?

Now you have correctly listed Indeterminate forms, of which #2 is none of.

So can you repost this so that it makes sense?
If I were to have such an example I would write it as [1/10^100]/[10^100] = [1/10^100]*[1/10^100]... so, the limit DNE

if the lim x->a f(x) approaches o and lim x->a p(x) approaches INF =>the lim DNE

1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
2. Lim x->a f(x)/p(x) = 0/INF, (DNE)
3. Lim x->a h(x)/p(x) = 1/INF, (Indeterminate form)
4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)

am I right (bc I feel I'm going to be more confused... )

ps: don't worry about the niceness while giving me advices, I appreciate much more the advices, not the way they are given

6. ## Re: Evaluate the limits

wrong post, sorry

7. ## Re: Evaluate the limits

Originally Posted by dokrbb
If I were to have such an example I would write it as [1/10^100]/[10^100] = [1/10^100]*[1/10^100]... so, the limit DNE

if the lim x->a f(x) approaches o and lim x->a p(x) approaches INF =>the lim DNE

1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
2. Lim x->a f(x)/p(x) = 0/INF, (DNE)
3. Lim x->a h(x)/p(x) = 1/INF, (Indeterminate form)
4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)

am I right (bc I feel I'm going to be more confused... )
so...?

8. ## Re: Evaluate the limits

Originally Posted by dokrbb
so...?
It truly troubles me that you are seemingly in a calculus course but know so very little about limits.

LEARN all of those forms. Then complete rework what you posted, because most of your answers are completely wrong.

9. ## Re: Evaluate the limits

Thanks Plato, I got them:

1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
2. Lim x->a f(x)/p(x) = 0/INF = 0 bc as f(x) approaches 0, p(x) becomes large
3. Lim x->a h(x)/p(x) = 1/INF = 0 since when h(x) approaches a finite nr p(x) becomes large
4. Lim x->a p(x)/f(x) = INF/0, DNE (I think we would need to evaluate limit from the right and from the left in order to be able to determine the lim)
5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)

10. ## Re: Evaluate the limits

Originally Posted by dokrbb
I got them:
1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
2. Lim x->a f(x)/p(x) = 0/INF = 0 bc as f(x) approaches 0, p(x) becomes large
3. Lim x->a h(x)/p(x) = 1/INF = 0 since when h(x) approaches a finite nr p(x) becomes large
4. Lim x->a p(x)/f(x) = INF/0, DNE (I think we would need to evaluate limit from the right and from the left in order to be able to determine the lim)
5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)
And that looks much better.

11. ## Re: Evaluate the limits

The only remark is the following.

Originally Posted by dokrbb
4. Lim x->a p(x)/f(x) = INF/0, DNE (I think we would need to evaluate limit from the right and from the left in order to be able to determine the lim)
I am not sure if INF is included in DNE or not. The limit $\lim_{x\to a}p(x)/f(x)$ can be $\pm\infty$ or it may not exist. The former happens when f(x) does not change sign in some neighborhood of a, and the latter happens when there are both positive and negative values f(x) when x is arbitrarily close to a: then p(x) / f(x) changes sign as x tends to a while its absolute value continues to grow.

12. ## Re: Evaluate the limits

Originally Posted by emakarov
The only remark is the following.

I am not sure if INF is included in DNE or not. The limit $\lim_{x\to a}p(x)/f(x)$ can be $\pm\infty$ or it may not exist. The former happens when f(x) does not change sign in some neighborhood of a, and the latter happens when there are both positive and negative values f(x) when x is arbitrarily close to a: then p(x) / f(x) changes sign as x tends to a while its absolute value continues to grow.
well, yes but since I'm allowed to give separate answers: whether $\lim_{x\to a}p(x)/f(x)$ is a) $\p\infty$ ,b) $-\infty$ , c) DNE or don't have enough information to evaluate it, I considered the more appropriate answer in this case is c)

and thanks a lot

13. ## Re: Evaluate the limits

Originally Posted by dokrbb
well, yes but since I'm allowed to give separate answers: whether $\lim_{x\to a}p(x)/f(x)$ is a) $\infty$ ,b) $-\infty$ , c) DNE or don't have enough information to evaluate it, I considered the more appropriate answer in this case is c)
When you say that the answer is c), it means that the limit does not exists for all function p(x) and f(x) under given assumptions. And this is not true: for some p(x) and f(x), the limit is +∞ or -∞. Therefore, I think the correct answer is that there is not enough information.

14. ## Re: Evaluate the limits

Originally Posted by emakarov
When you say that the answer is c), it means that the limit does not exists for all function p(x) and f(x) under given assumptions. And this is not true: for some p(x) and f(x), the limit is +∞ or -∞. Therefore, I think the correct answer is that there is not enough information.
Frankly I am confused at this point, myself.
In the OP we are given that ${\lim _{x \to a}}f(x) = 0\quad \& \quad {\lim _{x \to a}}p(x) = \infty$.

We are also told that "c) DNE or don't have enough information to evaluate it" Also "I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we don’t have enough info to determine the limit[I]."
I did not think that "that there is not enough information" was an option.

15. ## Re: Evaluate the limits

Originally Posted by Plato
Frankly I am confused at this point, myself.
In the OP we are given that ${\lim _{x \to a}}f(x) = 0\quad \& \quad {\lim _{x \to a}}p(x) = \infty$.

We are also told that "c) DNE or don't have enough information to evaluate it" Also "I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we don’t have enough info to determine the limit[I]."
I did not think that "that there is not enough information" was an option.
the confusing statement is that for c) the both answers are mentioned (1)DNE or 2) not enough info); therefore, whether I consider the situation 1) or 2), I have to chose the same c) answer,

sorry, but that's how the options were stated, and I checked the answers - it was c) the correct answer