Here we will prove properly that our intuitive calculations yielded the correct answer. We will use the usual method: factor out the dominant terms as we guessed them.

It will be easier to handle the resulting fractions separately.

In the last calculation, the fact that x²/4^x goes to 0 is proved just like in the third fraction, the calculation uses l'Hospital's rule twice in a row.

Now we can put it together.

The last equality follows similarly to the third fraction above, it is established using two l'Hospitals.

As a bonus we will prove that our intuitive simplification of roots was correct, that is, we will show that the comparison of behavior was correct (see the section on order of functions). In fact, we will prove it only for the first root:

When you write the appropriate limit for the second root comparison, you will see that we already did exactly the right calculations above.