1. Evaluating Limits

Hi, I'm new here and this is my first post but I'm having trouble with one of my homework problems. I have the function:

g(x) = (a(x+b)(x-c)(x+d)) / (e(x-f)(x-b)(x-c)), where a, b, c, d, e, and f are all unique, positive real numbers.

a) At what values of x=h does the limit g(x) = -∞ or +∞ as x->h?

b) Evaluate the limit of g(x) as x->c.

c) Does the limit g(x) as x->∞ exist?

I'm completely stuck and unsure where to start...

2. Re: Evaluating Limits

Originally Posted by daveMathews48
g(x) = (a(x+b)(x-c)(x+d)) / (e(x-f)(x-b)(x-c)), where a, b, c, d, e, and f are all unique, positive real numbers.
a) At what values of x=h does the limit g(x) = -∞ or +∞ as x->h?

b) Evaluate the limit of g(x) as x->c.

c) Does the limit g(x) as x->∞ exist?

I'm completely stuck and unsure where to start...
−∞−∞
This is very good question to teach you about limits.

The answer to a) is $\displaystyle f~\&~b$. Now you reply giving the reasons.
Then maybe we can do more.

3. Re: Evaluating Limits

If I cancel out the (x-c) from both the top and bottom then set the new denominator e(x-f)(x-b) to 0? Wouldn't that leave me with e, f, and b?

4. Re: Evaluating Limits

Originally Posted by daveMathews48
If I cancel out the (x-c) from both the top and bottom then set the new denominator e(x-f)(x-b) to 0? Wouldn't that leave me with e, f, and b?
No it does not. Because $\displaystyle e$ is a positive number.

5. Re: Evaluating Limits

Ah that's right, thanks. So for part b with (x-c) cancelled out, how do I evaluate that limit?

6. Re: Evaluating Limits

For part b) the answer is a fraction. The numerator of which is $\displaystyle a(c+b)(c+d)$.
What is its denominator?

7. Re: Evaluating Limits

Are you substituting c for x? That would make the denominator e(c-f)(c-b)?

8. Re: Evaluating Limits

Also, I was thinking about part c. Doesn't lim g(x) as x-> ∞ look for values where x is really large? If I expand the numerator and denominator I get:

(a(x^2+bx+dx+bd)) / (e(x^2-bx-fx+bf)), which for large x leaves me with ax^2/ex^2? Does that mean the limit exists and is a/e?

9. Re: Evaluating Limits

Originally Posted by daveMathews48
Are you substituting c for x? That would make the denominator e(c-f)(c-b)?
In limits NEVER JUST SUBSTITUTE!. Ask what happens near the limiting value?
If $\displaystyle x$ is near $\displaystyle c$ then $\displaystyle (x-b)$ is near $\displaystyle c-b$.

Now for the last part. We play the 'back-of-the-book" game.
You look up the answer. You see: "the limit exists and equals $\displaystyle \frac{a}{e}$".