# Why does Lim as x approaches infinity of x/(x-9) = 1?

1. Feb 8, 2012

### LOLItsAJ

1. The problem statement, all variables and given/known data

Limit as x approaches infinity of $x/x-9$

2. Relevant equations

None

3. The attempt at a solution

I know the indeterminate form infinity/infinity happens. I don't know how to fix it, but I'm assuming it's quite simple...

2. Feb 8, 2012

### Xyius

When x is very very large, say 6 billion. The numbers 6 billion and 6 billion minus 9 are essentially the same. As x gets larger and larger, this discrepancy diminishes. The value the ratio approaches is 1.

If you wanted to be a little more rigorous, you can divide the numerator and denominator by 1/x then take the limit and see what happens. It should pop right out.

3. Feb 9, 2012

### Totalderiv

A nice trick for these types of problems would be to divide everything by x;

$$\lim_{x\rightarrow\infty} \frac{x/x}{(x/x)-(9/x)}$$

Then simplify and you should be able to figure it out from there

4. Feb 9, 2012

### Voodoo583

Using L'Hopitals rule for evaluating indeterminate forms, just differentiate both the numerator and the denominator separately to produce 1/1. Xyius explains well intuitively why the result is what it is.