Fall 2023 MNC: Take it to the limit with numerical, graphical, algebraic representations

The video begins with Curtis Brown enthusiastically welcoming viewers to another season of Monday Night Calculus. He is excited about the consistent schedule this season, with all events happening on Monday nights. Curtis introduces his colleagues, Steve Kokoska and Tom Dick, and encourages students to ask questions and seek extra credit opportunities. The session aims to be interactive, with a focus on limits using numerical, graphical, and algebraic approaches. Curtis highlights the importance of the chat for questions and acknowledges that the upcoming discussions are not proofs but rather estimations of limits.

Curtis and his team delve into the specifics of estimating limits using tables of values and graphical representations. They discuss the behavior of functions as x approaches zero, using the example of x squared times the log of x. Curtis shares his thought process as he examines the function, considering the competition between x squared approaching zero and the log function approaching negative infinity. The use of technology, specifically Mathematica, is introduced to illustrate the function's graph and how it crosses the x-axis. The conversation transitions to a new technology platform, with some initial technical difficulties acknowledged.

The discussion shifts to the limit of the composition function, log of sine of x, as x approaches zero from the right. Curtis uses trigonometry to predict the behavior of the sine function and expects the composition to approach negative infinity. This expectation is confirmed through a table of values and a graph. Open questions are posed for further exploration, such as the function's behavior as x increases without bound and the possibility of vertical asymptotes. Curtis then attempts another limit problem, involving a difference of functions, and shares his reasoning process before looking at the table and graph.

Tom takes over to demonstrate the use of a TI-84 calculator to analyze the limits discussed by Steve. He walks through the process of entering functions, graphing, and using the table feature to investigate the behavior of function values as x approaches zero. Tom emphasizes that these are not proofs but rather estimations. He also addresses a chat question about changing a subtraction sign to a plus sign and how to approach that on a calculator. The demonstration includes adjusting the table settings for a better understanding of limit behavior and using the ask function for more precise inputs.

Tom continues to explore the behavior of functions near zero using the calculator. He examines the function x squared times the natural log of x and uses the table feature to find function values as x approaches zero from the right. The values indicate that the function is approaching zero. Tom also discusses the limitations of the table feature and how to work around them by manually inputting values. He then moves on to another function, the natural log of the sine of x, and examines its behavior near zero, noting the slow decrease towards zero and the potential for a vertical asymptote.

Curtis and Tom present a series of challenging limit problems, encouraging viewers to think critically about the properties of limits. They discuss the existence of limits for piecewise functions and the conditions under which limit properties can be applied. Curtis presents a problem involving the product of two functions as x approaches one and emphasizes the need to examine one-sided limits when the intuitive approach fails. The problem is solved by considering the left and right limits separately, leading to the conclusion that the overall limit exists and is zero. Tom then explores another limit involving a composition of functions, highlighting the importance of function continuity.

Curtis and Tom tackle a series of limit problems using both algebraic and graphical methods. They discuss the importance of notational fluency and carrying the limit expression through the solution process. Curtis solves a limit problem algebraically by rationalizing the denominator and finds that the limit is two-thirds. Tom then uses the TI-84 to graph the function and verify the solution. They also discuss the domain of the function and the presence of a vertical asymptote at x equals minus one. Tom demonstrates how to enter piecewise functions into the calculator and explores the combination of two functions to analyze their product and limit behavior as x approaches one.

Curtis concludes the session by thanking everyone for their participation and expressing his gratitude for the engagement with the overtime problems. He encourages teachers to request professional development certificates via email and reminds them to check for the student document and solutions to be posted in the coming weeks. Curtis also mentions that the next session will be held on October 2nd and encourages students and teachers to join for further learning and discussion.