Problem: Evaluate the following limit (if it exists)
Solution: First we put in infinity to see what kind of expression we have:
No question here, we should look into the box indeterminate product. The method calls for changing the product into a fraction. Which part will we "put under"? Putting down the arctan part would result in something rather complicated, so there is little doubt that putting down the n is a better choice:
Just as expected, we obtained an indeterminate ratio, so we should switch to functions and apply the l'Hospital rule.
We used the l'Hospital rule twice, but the second time we could have tried something else. We faced a ratio of polynomials, and in the appropriate box we are advised to determine the dominant terms and then factor them out. Here the dominant terms in both the numerator and the denominator are x2, so it is easier to just cancel it:
