Understanding Roots, Limits, and Continuity in Single-Variable Calculus
This chapter introduces key concepts in single-variable calculus, focusing on roots (or zeros) and limits. A root of a function f(x) is defined as a value a for which f(a) = 0. We explore conditions for a limit to exist and provide examples illustrating behaviors around certain values, including when limits do not exist. Additionally, we define continuity and provide insights into the implications of switching limits and functions. The chapter concludes with a discussion on the bounded nature of sine and cosine functions and the importance of continuity for the Intermediate Value Theorem.
Understanding Roots, Limits, and Continuity in Single-Variable Calculus
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Presentation Transcript
Chapter 1 Summary Math 1231: Single-Variable Calculus
Roots or Zeros: Definition • Given a function f(x), we call x = a is a ROOT or ZERO of f(x) if f(a) = 0. • Given a equation f(x)=0, we call x = a is a root or zero or SOLUTION of the equation if f(a) = 0. Example: x = π/2 is a root of cos(x); Example: x = 1 is a root of x2 + x = x + 1.
Limit • limx->af(x) = L exists if, for ANY sequence of x values that approach a, f(x) always becomes sufficiently close to L. • Example: limx->2 x2 = 4 means, for any sequence of x values, • x = 2.1, 2.01, 2.001, 2.0001, … • x = 1.9, 1.99, 1.999, 1.9999, … • x = 2.2, 1.8, 2.02, 1.98, 2.002, 1.998, … x2 always sufficiently approaches 4.
When Does Limit Not Exist? • If f(x) approaches two different values when x approaches a along two different sequences, then f(x) does not have a limit. • Example: f(x) = sin(1/x) does not have a limit when x approaches 0, because by taking x to be • x = 1/π, 1/2π, 1/3π, …, f(x) 0 • x = 1/0.5π, 1/2.5π, 1/4.5π, …, f(x)1
More Examples For any point x = a, limx-> af(x) does not exist. So f(x) is discontinuous everywhere. limx-> af(x) does not exist for all the points except x = 0. Furthermore, f(x) is continuous only at x = 0.
Continuity f(x) = f1(x) f(x) = f2(x) f(x) = f3(x) 2 4
Continuity f(x)iscontinuousataif limx->af(x)=f(a) Continuityimplies we can switch the order of limandf Example: Suppose that limx->1 g(x) = π, find limx->1 cos(g(x)).
Miscellaneous • sine and cosine functions are bounded by -1 and 1.
Miscellaneous • I.V.T. is only valid when f(x) is continuous.
