Limit Proofs

[Djohnst76]

Hi all,

This is an Extra Credit Problem, and I have no idea where to begin. Any help would be appreciated.

Problem A. Let a, b, and c be constant real numbers such that lim x→c [f(x) + g(x)] = a and lim x→c [f(x)−g(x)] = b.

Find lim x→c [f(x)g(x)] in terms of a and b.

 

[JeffM]

If you knew what [MATH]\lim_{x \rightarrow c}f(x)[/MATH] equaled,

could you solve the problem?

But how do you find that out? Basically, you need to know and understand these basic laws of limits.

Assuming the limits of f(x) and g(x) at c are real numbers, then

[MATH]\lim_{x \rightarrow c} \{ f(x) + g(x) \} = \{ \lim_{x \rightarrow c}f(x) \} + \{ \lim_{x \rightarrow c} g(x) \};[/MATH]
[MATH]\lim_{x \rightarrow c} \{ f(x) - g(x) \} = \{ \lim_{x \rightarrow c}f(x) \} - \{ \lim_{x \rightarrow c} g(x) \};[/MATH]
[MATH]\lim_{x \rightarrow c} \{ f(x) * g(x) \} = \{ \lim_{x \rightarrow c}f(x) \} * \{ \lim_{x \rightarrow c} g(x) \}; \text { and}[/MATH]
[MATH]\lim_{x \rightarrow c} g(x) \ne 0 \implies \left \{ \lim_{x \rightarrow c} \dfrac{f(x)}{g(x)} \right \} = \dfrac{\displaystyle \lim_{x \rightarrow c}f(x)}{ \displaystyle \lim_{x \rightarrow c} g(x)}.[/MATH]
Now see if you can use the first two of those to find what

[MATH]\lim_{x \rightarrow c}f(x)[/MATH] equals?

Give it a try. Show what you get. If this hint is not enough, show your work and we shall give another hint.

 

[pka]

This is an Extra Credit Problem, and I have no idea where to begin. Any help would be appreciated.
Problem A. Let a, b, and c be constant real numbers such that lim x→c [f(x) + g(x)] = a and lim x→c [f(x)−g(x)] = b.
Find lim x→c [f(x)g(x)] in terms of a and b.

Because this is an extra credit question, it is unfair to give you the answer.
However, I would expect that you must show work to get full credit.
So playing the back-of-the-book game, look-up the answer you will see \(\dfrac{a^2-b^2}{4}\). HINT \([f\pm g]^2=f^2{\pm\bf{2fg}}+g^2\)

 

[Steven G]

I'd add, subtract and divide by 2 twice to get somewhere.

 

[Steven G]

BTW, thank you for telling us this is for extra credit.

 

[Djohnst76]

Once I went back and reviewed the limit laws this is what I came up with see attached, it looks like other replies agree.

See attached.

Thanks for the replies, this is for the the first section of Calculus and I am still getting used to some of it.

 

[Steven G]

Did you get anywhere with the problem? Please share your work with us.

 

[Djohnst76]

[/ISPOILER]

Did you get anywhere with the problem? Please share your work with us.

I did, but it is awaiting moderation. There was a similar problem in the text, with values for a and b. After reviewing the limit laws and trying that problem I put the two equations into a system and solved from there. Hopefully, my work will show up soon.

I am not familiar with Latex yet, is there a way to paste Word equations into this thread? Or is it easiest to just post pics of future problems?

 

[Steven G]

Once I went back and reviewed the limit laws this is what I came up with see attached, it looks like other replies agree.

See attached.

Thanks for the replies, this is for the the first section of Calculus and I am still getting used to some of it.

Nicely done.