Unit 1B

Limits

Lesson 2: Algebraic Limits

Much of the credit for the discovery of calculus goes to two individuals. In the mid 1600s, both Isaac Newton and Gottfried Leibniz independently developed and codified the basic ideas of calculus. However, going as far back as Ancient Greece, and as distant as India and China, mathematicians throughout history were slowly laying the foundations of calculus.

As you will see in subsequent chapters, much of the foundation of calculus is built on limits. Prior to the invention of the mechanical and electronic calculators, a period of time that corresponds to most of the history of calculus, mathematicians worked almost exclusively by pencil and paper. As a result, it was, and still is, extremely important to understand the numeric and algebraic methods of determining limits.

The previous Lesson began with a look at the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»9«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math». The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»3«/mn»«/mrow»«/munder»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»9«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» was determined graphically, using a graph that was provided. What if, like early mathematicians, you didn’t have the graph of the function readily available? What methods could you use to evaluate a limit or determine if it even exists?

In addition to graphical analysis, numerical approximation and algebraic analysis can be employed to evaluate limits.

Numerical Approximation

This method involves substituting values closer and closer to the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-value the limit is approaching and calculating the corresponding values of the function. A substantial amount of calculating is required, which can be tedious without a calculator. However, it is a very useful method and should be understood.