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,

,

or

.

Clearly and . Therefore, by the Sandwich Theorem, we can conclude that .

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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