Chapter 1, Section 1.1: Limit of a Function , remixed by Jeff Eldridge from work by Dale Hoffman , is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License . © Mathispower 4u.
62 contemporary calculus 1.1The Limit of a FunctionCalculus has been called the study of continuous change, and the limitis the basic concept that allows us to describe and analyze such change.An understanding of limits is necessary to understand derivatives,integrals and other fundamental topics of calculus.The Idea (Informally)The limit of a function at a point describes the behavior of the functionwhen the variable is near—but does not equal—a specified number (seemargin figure). If the values off(x)get closer and closer—as close aswe want—to one numberLas we take values ofxvery close to (butnot equal to) a numberc, thenThe symbol→means “approaches” or“gets very close to.”we say: “the limit off(x), asxapproachesc, isL”and we write: limx→cf(x) =LIt is very important to note that:f(c)is a single number that describes the behavior (value)offatthe pointx=cwhile:limx→cf(x)is a single number that describes the behavioroffnear,but not atthe pointx=cIf we have a graph of the functionf(x)nearx=c, then it is usuallyeasy to determine limx→cf(x).Example1.Use the graph ofy=f(x)given in the margin to determinethe following limits:(a)limx→1f(x)(b)limx→2f(x)(c)limx→3f(x)(d)limx→4f(x)
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