How did we know that the indeterminate ratios went all to zero? They should when we divide a non-dominant term by a dominant one (that's what dominance is all about), mathematically we do it most often using l'Hospital's rule. However, sometimes we are better off with some tricks.

This is definitely true about the first fraction, which admittedly could be done using l'Hospital, but used 13 times. If you feel like it, you are welcome. Here we will show a little dirty trick.

The second fraction is a straightforward l'Hospital.

The third fraction is easily done by algebra; here the l'Hospital rule would not help at all.

The fourth fraction would again require 13 l'Hospitals, so we use a similar trick as with the first one.