How does the proof of L'Hopital's rule show that the limit is equal to A?

[modnarandom]

I've been reading over the l'Hopital's rule proof here: http://www.math.uga.edu/~pete/2400diffmisc.pdf

For the first case of the proof, how does f(x)/g(x) <α imply the limit as x -> a is also A? I understand that α is an arbitrary number greater than A, which means that the limit is less than or equal to A. But I'm not sure why equals A. Does it have something to do with t?

Thank you!

 

[Robert1986]

I've been reading over the l'Hopital's rule proof here: http://www.math.uga.edu/~pete/2400diffmisc.pdf

For the first case of the proof, how does f(x)/g(x) <α imply the limit as x -> a is also A? I understand that α is an arbitrary number greater than A, which means that the limit is less than or equal to A. But I'm not sure why equals A. Does it have something to do with t?

Thank you!

You also have to have Step 2, as well. That shows that we can "trap" f(x)/g(x) in some arbitrary interval about A.