1. ## Limits at infinity

Finding the limit of (9x + 5)/(7x + 9) as x approaches ∞.

So, I have to divide the numerator and denominator by the highest power of x in the denominator? If so, that would be 1. Therefore, I would have to divide both the numerator and denominator by x/1?

Is that correct?

Then I would get (9 + 5/x)/(7 + 9/x). Where would I go from there?

Okay, I got it. 5/x and 9/x become zero, so the answer is 9/7.

Do you always divide by the highest power of x in the denominator, and never the numerator?

2. Originally Posted by ImaCowOK
Finding the limit of (9x + 5)/(7x + 9) as x approaches ∞.
Do you always divide by the highest power of x in the denominator, and never the numerator?
None of us likes to answer a question that contains always
These are rules of thumb and nothing more.
If you have the ratio of two polynomials that is a correct rule of thumb.

3. I know we are not supposed to generalize in math, I've heard it said many times already. I would like to know the rules, if when solving these I will have to use the highest power of x in the denominator. If I have one with x² in the numerator, but the highest power in the denominator is just x - do I divide by x or x²?

4. The point is: there are no hard and fast rules.
If we have two polynomials, say $\dfrac{x^2+x+1}{x+1}$ then divide by $x$.
The limit is infinity.

5. for functions formed by the ratio of polynomials, there are three cases ...

degree of numerator > degree of denominator ... as $x \to \infty$ the value of the rational function increases w/o bound.

degree of numerator < degree of denominator ... as $x \to \infty$ the value of the rational function approaches 0.

degree of numerator = degree of denominator ... as $x \to \infty$ the value of the rational function approaches the ratio of the leading coefficients.

6. Originally Posted by Plato
there are no hard and fast rules.
I know that.

You resolved that last question: I would divide by x in that case.