Learning Outcomes

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


  • 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

  1. Evaluate the limit of $\displaystyle\lim_{x\to 1/2}\frac{2x^2+3x-2}{2x-1}$
  2. 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 factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction.


Videos and Practice Problems of Selected Topics

  • Evaluating limits using limit laws
  1. $\rhd$  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?
  2. $\rhd$ 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 $\displaystyle\lim_{x\to 0}\dfrac{h(x)}{g(x)}$.
  3. $\rhd$ 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 $\displaystyle\lim_{x\to 1}(f(x)g(x))$.
  4. * Practice:  Limits of combined functions: sums and differences. (4 problems)
  5. * Practice:  Limits of combined functions: products and quotients. (4 problems)
  6. $\rhd$ 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 $\displaystyle\lim_{x\to -3}h(f(x))$.
  7. * Practice: Limits of composite functions. (4 problems)
  • Evaluating limits using algebraic manipulations
  1. $\rhd$ Limits by factoring (5:44) Find $\displaystyle\lim_{x\to 2}\dfrac{x^2+x-6}{x-2}$.
  2. * Practice: Limits by factoring. (4 problems)
  3. $\rhd$ Limits by rationalizing (9:31) Find $\displaystyle\lim_{x\to -1}\dfrac{x+1}{\sqrt{x+5}-2}$.
  4. * Practice: Limits using conjugates. (4 problems)