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

[LOLItsAJ]

Homework Statement

Limit as x approaches infinity of x/x-9

Homework Equations

None

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...

 

[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.

 

[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

 

[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.