Proving Limits

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March 5th 2010, 01:20 AM
Monkee

Proving Limits

I have never felt so lost in my life, so far.(Doh) Hopefully someone can help.

The section is on "The Precise Definition Of A Limit"

Suppose that $\lim\ _{x \to a}\ f(x)=\infty$ and $\lim\ _{x \to a}\ g(x)= c$ , where $c$ is a real number. Prove each statement.

$(a)$ $\lim_{x \to a}[f(x)+g(x)]=\infty$

$(b)$ $\lim_{x \to a}[f(x)g(x)]=\infty$ if $c>0$

$(b)$ $\lim_{x \to a}[f(x)g(x)]=-\infty$ if $c<0$

Am I supposed to use the triangle inequality? I think that if someone can help with problem (a), then I should be able to figure out the rest. Thanks!
March 5th 2010, 01:23 AM
Prove It

Quote:

Originally Posted by Monkee

I have never felt so lost in my life, so far.(Doh) Hopefully someone can help.

The section is on "The Precise Definition Of A Limit"

Suppose that $\lim\ _{x \to a}\ f(x)=\infty$ and $\lim\ _{x \to a}\ g(x)= c$ , where $c$ is a real number. Prove each statement.

$(a)$ $\lim_{x \to a}[f(x)+g(x)]=\infty$

$(b)$ $\lim_{x \to a}[f(x)g(x)]=c$ if $c>0$

$(b)$ $\lim_{x \to a}[f(x)g(x)]=-\infty$ if $c<0$

Am I supposed to use the triangle inequality? I think that if someone can help with problem (a), then I should be able to figure out the rest. Thanks!

What are you allowed to take for granted?

It is well known that the limit of a sum is the same as the sum of the limits.

It is also well known that the limit of a product is the same as the product of the limits.

Also, b) is wrong. It should be $\infty$ , not $c$ .
March 5th 2010, 01:38 AM
Monkee

Hi Prove It, thanks for the quick reply and catching my typo. I am not sure what you mean by what is taken for granted but would it be correct if I said that $<br /> \lim_{x \to a}[f(x)+g(x)]=\infty+c<br />$ ?

Is $\frac{\infty}{2}>0$ somewhere in the proof?
March 5th 2010, 01:42 AM
Prove It

Quote:

Originally Posted by Monkee

Hi Prove It, thanks for the quick reply and catching my typo. I am not sure what you mean by what is taken for granted but would it be correct if I said that $<br /> \lim_{x \to a}[f(x)+g(x)]=\infty+c<br />$ ?

What I mean is, I'm not sure if you're allowed to use these results in your class without first proving them.

And yes, what you have written is correct (at least logically, not so much mathematically, since $\infty$ is not a number...)
March 5th 2010, 01:51 AM
Monkee

Ahh, I see what you mean now. Yes we are allowd to use the results. All we need to do is prove them.
March 5th 2010, 01:56 AM
Prove It

Then I suggest you read the following proofs about the properties of limits.

Pauls Online Notes : Calculus I - Proof of Various Limit Properties
March 5th 2010, 02:19 AM
Monkee

Quote:

Originally Posted by Prove It

Then I suggest you read the following proofs about the properties of limits.

Pauls Online Notes : Calculus I - Proof of Various Limit Properties

Thanks! That is very similar to my book but I like the way it is layed out on the website. One more question before I try to tackle this: Does the given " $\varepsilon>0$ ", apply to my problem as well?
March 5th 2010, 02:38 AM
Prove It

Quote:

Originally Posted by Monkee

Thanks! That is very similar to my book but I like the way it is layed out on the website. One more question before I try to tackle this: Does the given " $\varepsilon>0$ ", apply to my problem as well?

Yes, the formal definition applies for any limit.