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Limits are a very powerful tool in mathematics and are used throughout calculus and beyond. The key idea is that a limit is what I like to call a \behavior operator". A limit will tell you the behavior of a function nearby a point.
Find an example of a function such that the limit exists at every x, but that has an in nite number of discontinuities. (You can describe the function and/or write a
Numbers and Functions The subject of this course is \functions of one real variable" so we begin by wondering what a real number \really" is, and then, in the next section, what a function is.
SECTION 1.5: PIECEWISE-DEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS LEARNING OBJECTIVES • Know how to evaluate and graph piecewise-defined functions. • Know how to evaluate and graph the greatest integer (or floor) function. • Preview limits and continuity from calculus. PART A: DISCUSSION
first really calculus-based notion: the limit. This is really a very intuitive concept, but it’s also kind of miraculous and lets us do some very powerful things. I’ll eventually write down a formal definition for a limit, but it’s not really important and I won’t ask you to use it: the important thing is that you understand the fundamental
Definition of Limit: “Let f be a function defined at each point of some open interval containing a, except possibly a itself. Then a number L is the limit of f (x) as x approaches a (or is the limit of f at a) if for every number ε > 0 there is a number δ > 0 such that if 0 , then < x − a < δ f (x)− L < ε”. In other words, if we ...
Theorem 1 (Limits and Bounds of Functions) Let f : X 7→R and suppose c is an accumu-lation point of X. If f has a limit at c, then there is a neighborhood Q of c and a real number m such that for all x ∈ Q∩X, |f(x)| ≤ m. By now, you might guess that there is the strong connection between limits of sequences and functions.
