Learning Outcomes

  1. Use tables to estimate the limit of a function or to identify when the limit does not exist
  2. Use graphs to estimate the limit of a function or to identify when the limit does not exist
  3. One-sided and two-sided limits and their relationship

Textbook

  • Chapter 2.2  The limit of a Function  

Textbook Assignment

  • p. 154:   30-33 all, 35, 38, 42

WeBWorK Assginment

  • Limits-Introduction

Exit problems of the session

  1. Use a table to estimate the limit of $\displaystyle\lim_{x\to 0}\frac{\tan x}{2x}$.
  2. Use a graph to estimate limits: Textbook pp. 157: 59-64

 

Key Concepts

  • A table of values or graph may be used to estimate a limit.
  • If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
  • If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
  • We may use limits to describe infinite behavior of a function at a point.

 

 

Videos and Practice Problems of Selected Topics

  • Estimating the limit using graphs
  1. $\rhd$  Estimating limit values from graphs (6:00) Two graphs are given. Find $\displaystyle\lim_{x\to 6}f(x)$, $\displaystyle\lim_{x\to 4}f(x)$ and $\displaystyle\lim_{x\to 2}f(x)$, $\displaystyle\lim_{x\to 5}g(x)$, $\displaystyle\lim_{x\to 7}g(x)$ and $\displaystyle\lim_{x\to 1}g(x)$.
  2. $\rhd$ Unbounded limits (2:31) A discussion on $\displaystyle\lim_{x\to 0}\dfrac{1}{x^2}$ and $\displaystyle\lim_{x\to 0}\dfrac{1}{x}$ using graphs.
  3. $\rhd$ One-sided limits (9:10) Three graphs are given and several one-sided limits are estimated.
  4. * Practice: Analyze the graph of a function to find the limit. (5 problems with a guiding text)
  5. * Practice: Estimating limit values from graphs. (6 problems with a guiding text)
  6. * Practice: Estimating limit values from graphs. (4 problems)
  7. * Practice: One-sided limits. (4 problems)
  • Estimating the limit using tables
  1. $\rhd$ Approximating limits using tables (4:26) Use tables to estimate $\displaystyle\lim_{x\to 3}\dfrac{x^3-3x^2}{5x-15}$.
  2. $\rhd$ Estimating tables from limits (3:23) Selected values of $g(x)$ are given to estimate $\displaystyle\lim_{x\to 5}g(x)$.
  3. * Practice: Using tables to approximate limit values. (5 problems with a guiding text)
  4. * Practice: Creating tables for approximating limits. (4 problems)