SUNY Geneseo Department of Mathematics
Math 221 02
Fall 2020
Prof. Doug Baldwin
Some expressions involving limits cannot at first be evaluated using just the limit laws, usually because attempts to use those laws result in a division of the form 0 / 0. This discussion consolidates your ability to use algebra to simplify such limit expressions, as illustrated in the “Additional Limit Evaluation Techniques” subsection of section 2.3 in our textbook.
Respond to one or more of the following by class time on Thursday, September 10, and one or more additional prompts by class time on Friday, September 11. (In other words, this is a 2-day discussion, but I want you to keep engaging with it through that whole time.) Each prompt offers you a limit, and it would eventually be nice to actually find that limit. But you don’t actually have to do the whole process of finding a limit in order to “respond” to it — you could just as well suggest a technique without actually carrying it out, ask a question, extend someone else’s partial solution, correct a mistake you spotted, etc. We will present — either because you did them here or because we do them in class — complete solutions to some or all of these in the class notes for Thursday and Friday.
\[\lim_{x \to 1} \frac{x^2 - 2x + 1}{x - 1}\] \[\lim_{\theta \to 0} \frac{2 \sin\theta}{\sin(2\theta)}\] \[\lim_{t \to 3} \frac{t - 3}{t^2 - 9}\] \[\lim_{x \to -1} \frac{x^2 + 2x + 1}{x^2 -2x -3}\] \[\lim_{y \to 2} \frac{\sqrt{y-1} - 1}{y - 2}\] \[\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}\] \[\lim_{z \to 0} \frac{ \frac{3}{(1+z)^2} - \frac{3}{1+z}}{z}\]