Explain why one of the following limits can use L’Hospital’s rule and the other cannot.Evaluate both limits.

Question

a. lim 𝑥→∞ 𝑥tan^−1 (1/𝑥)

b. lim 𝑥→∞ (tan^−1𝑥)(cot^−1𝑥)

Answer 1

The L'Hospital rule works if the result is 0/0 or infinity/infinity

In the first case, the result is infinity/infinity

In the second case, the tan^-1 function goes to .5pi & the cot^-1 function goes to zero. Giving limit = .5pi/0