Learning Outcomes

- Use the limit laws to evaluate the limit of a function
- Evaluate the limit of a function by factoring or by using conjugates
- Evaluate the limit of a piecewise-defined function

**Textbook**

- Chapter 2.3 The Limit Laws

**Textbook Assignment**

- p. 176: 83-101 odd

**WeBWorK** **Assginment**

- Limits-Analytic
- Limits-One-Sided
- Limits-Limit Properties

**Exit problems ****of the session **

- Evaluate the limit of $\displaystyle\lim_{x\to 1/2}\frac{2x^2+3x-2}{2x-1}$
- Assuming that $\displaystyle\lim_{x\to 3}f(x)=4$, $\displaystyle\lim_{x\to 3}g(x)=-5$, evaluate $\displaystyle\lim_{x\to 3}\frac{f^2(x)-1}{g(x)}$

** Key Concepts**

- The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time.
- You can evaluate the limit of a function by
, by*factoring and canceling*, or by*multiplying by a conjugate*.*simplifying a complex fraction*

#### Videos and Practice Problems of Selected Topics

**Evaluating limits using limit laws**

- $\rhd$ Limit properties (5:07) What is the limit of the sum of two functions? Difference? What about the product? Division? A function rasised to a number?
- $\rhd$ Limits of combined functions (4:08) Given the graphs of $f(x)$ and $h(x)$, find $\displaystyle\lim_{x\to 0}(f(x)h(x))$. Then, given the graphs of $g(x)$ and $h(x)$, find $\displaystyle\lim_{x\to 0}\dfrac{h(x)}{g(x)}$.
- $\rhd$ Limits of combined functions: piecewise functions (4:12) The graphs of two piecewise functions, $f(x)$ and $g(x)$, are given. Find $\displaystyle\lim_{x\to -2}(f(x)+g(x))$, $\displaystyle\lim_{x\to 1}(f(x)+g(x))$ and $\displaystyle\lim_{x\to 1}(f(x)g(x))$.
- * Practice: Limits of combined functions: sums and differences. (4 problems)
- * Practice: Limits of combined functions: products and quotients. (4 problems)
- $\rhd$ Limits of composite functions (5:11) Given the graphs of $g(x)$ and $h(x)$, find $\displaystyle\lim_{x\to 3}(g(h(x))$. Then, given the graphs of other functions $g(x)$ and $h(x)$, find $\displaystyle\lim_{x\to -1}h(g(x))$. Two more graphs are given for the functions $h(x)$ and $f(x)$ to find $\displaystyle\lim_{x\to -3}h(f(x))$.
- * Practice: Limits of composite functions. (4 problems)

**Evaluating limits using algebraic manipulations**

- $\rhd$ Limits by factoring (5:44) Find $\displaystyle\lim_{x\to 2}\dfrac{x^2+x-6}{x-2}$.
- * Practice: Limits by factoring. (4 problems)
- $\rhd$ Limits by rationalizing (9:31) Find $\displaystyle\lim_{x\to -1}\dfrac{x+1}{\sqrt{x+5}-2}$.
- * Practice: Limits using conjugates. (4 problems)