Limits

[alexedward]

If f and g are continuous functions with g(4) = 3 and lim as x->4 [f(x)]^2 / (f(x) - g(x)), find f(4).
Okay, so what I did was substitute f(4) in for f(x), and g(4) in for g(x). Since I know g(4) = 3, I changed g(4) to 3 in the limit equation. I know that I now need to rearrange the equation to solve for f(4), but I'm unsure of how to do that when there are two f(4) terms. Do I bring the entire denominator of the fraction over to the right-hand side and multiply it all by 12? Any other ideas? If it helps, the answer is supposed to be 6. Thank you for your time.

 

[mmm4444bot]

g(4) = 3 and lim as x->4 [f(x)]^2 / (f(x) - g(x)), find f(4)

I'm thinking that you forgot to tell us that the given limit equals 12.

I base my guess that it equals 12 on your question:

multiply it all by 12?

We could assign a symbol to represent the value of f(4).

c = f(4)

Now, if the given limit equals 12, we have the following.

c^2/(c - 3) = 12

And it seems that you're asking whether or not to multiply both sides by c - 3, followed by expanding the righthand side.

c^2 = 12(c - 3)

c^2 = 12c - 36

Yes, that is what I would do.

Solve this quadratic equation for c, and you'll have the value of f(4).

Cheers ~ Mark

 

[alexedward]

Oh wow, I'm sorry! Don't know why I didn't include that.
But thank you very much! It's so much easier replacing it with a variable.