What is the behavior of for positive and near , where , , , , and are positive? This Demonstration allows you to investigate.
If , then is bounded and the limit does not exist. If , then is unbounded. To see this, let and . Then, . What about the limit of along the algebraic curves , where ? If , then . If , then if and if . If , then can have limit 0, any positive number, or ∞, depending on . Let and be defined by , and . if , if , To see this, let , so , , where . Then, The results follow by considering the cases , , and . , as , and along any curve to the origin. Then, and . if , if , if . Then and if , if , and if .
