Learning Outcomes

- Use tables to estimate the limit of a function or to identify when the limit does not exist
- Use graphs to estimate the limit of a function or to identify when the limit does not exist
- 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 **

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

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**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**

- $\rhd$ 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 $\displaystyle\lim_{x\to 1}g(x)$.
- $\rhd$ 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.
- $\rhd$ One-sided limits (9:10) Three graphs are given and several one-sided limits are estimated.
- * Practice: Analyze the graph of a function to find the limit. (5 problems with a guiding text)
- * Practice: Estimating limit values from graphs. (6 problems with a guiding text)
- * Practice: Estimating limit values from graphs. (4 problems)
- * Practice: One-sided limits. (4 problems)

**Estimating the limit using tables**

- $\rhd$ Approximating limits using tables (4:26) Use tables to estimate $\displaystyle\lim_{x\to 3}\dfrac{x^3-3x^2}{5x-15}$.
- $\rhd$ Estimating tables from limits (3:23) Selected values of $g(x)$ are given to estimate $\displaystyle\lim_{x\to 5}g(x)$.
- * Practice: Using tables to approximate limit values. (5 problems with a guiding text)
- * Practice: Creating tables for approximating limits. (4 problems)