Help understanding infinite limits with L'Hopital's rule

[Draeus746]

Hello, I'm currently having some issues understanding infinite limits using the L'Hopital's rule. Currently both of the numerator and the denominator will keep breaking down and I don't understand what I'm supposed to do with the current derivatives I have gotten.

 

[Dr.Peterson]

Just keep going ... k times!

There's an unintentional hint to part of the answer in what I just wrote.

 

[Draeus746]

Oh, ok I think I see. It must be infinite because it seems as if the denominator can broken down k number of times.

 

[yoscar04]

After you finish following Dr. Peterson advice, you may check your result by expanding e^x.

 

[Dr.Peterson]

Oh, ok I think I see. It must be infinite because it seems as if the denominator can broken down k number of times.

In other words, after k repetitions, the numerator is still e^x, but the denominator has become a finite number, namely k!.