SUNY Geneseo Department of Mathematics
Math 221 02
Fall 2020
Prof. Doug Baldwin
This discussion helps develop ideas related to one-sided limits that are introduced in the “One-Sided Limits” subsection of section 2.2 in the textbook.
Post at least one response to at least one of the following by class time on Monday, September 14.
Consider the function defined by
\[f\left(x\right)=\begin{cases} x & \mathrm{if\ } x < 1\\ x^2 & \mathrm{if\ } 1 < x < 2\\ x-2 & \mathrm{if\ } x \ge 2\end{cases}\]To check that you are comfortable with this style of function definition (and if you aren’t we can talk about it either here in the discussion or in class), what is f(0)? f(1.5)? f(2)? f(3)?
Notice that f(1) is undefined — none of pieces of the function applies when x = 1. But is limx → 1 f(x) defined? If so, what is it? How about the 1-sided limits as x approaches 1 from the left or from the right?
Is limx → 2 f(x) defined? If so, what is it? How about the 1-sided limits as x approaches 2 from the left or from the right?
The absolute value function is another place where 1-sided limits are often helpful.
What are
\[\lim_{t \to 1^-} \frac{t^2 + t - 2}{\left| t - 1 \right|}\]and
\[\lim_{t \to 1^+} \frac{t^2 + t - 2}{\left| t - 1 \right|}\]The other common source of 1-sided limit questions is functions that are defined piecewise (as in the first part of this discussion). Can absolute value be defined piecewise? If so, what does the definition look like?
Do you think one can use limit laws and algebraic methods to find 1-sided limits, similarly to how we used them for 2-sided limits? Why or why not?