One of my colleagues placed the following problem on an exam for his Calculus II course…
and was impressed by the variety of correct responses that he received. I thought it would be fun to discuss some of the different ways that this limit can be computed.
Method #5. Another geometric approach. The numbers and can be viewed as two sides of a right triangle with legs and and hypotenuse . Therefore, the length of the hypotenuse must be larger than the length of one leg but less than the sum of the lengths of the two legs. In other words,
Clearly and . Therefore, by the Sandwich Theorem, we can conclude that .