Limits and Their Properties

Limits and Their Properties Calculus Chapter 1

An Introduction to Limits Calculus 1.1

Calculus is… • The mathematics of change • Velocity • Acceleration • The mathematics of tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, etc., that enable scientists, engineers, and economists to model real-life situations. Calculus Chapter 1

Calculus is … • A limit machine with three stages • Precalculus • Limit process • Calculus formulation • Derivatives • Integrals Calculus Chapter 1

Tangent Line Problem • Except for vertical tangent lines, to find the tangent line you must simply find its slope • You already know a point Calculus Chapter 1

Secant line • Used to approximate slope of tangent line • A line through the point of tangency (P) and a second point on the curve (Q) Calculus Chapter 1

Slope of secant line • As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line. •  slope of the tangent line is said to be the limit of the slope of the secant line Calculus Chapter 1

Limit • If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x) as x approaches c is L Calculus Chapter 1

Example • What happens at x = 2? • To get an idea, look at values close to 2 from the left and right Calculus Chapter 1

Important to remember • The existence or nonexistence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x approaches c. Calculus Chapter 1

Example Calculus Chapter 1

Limits that fail to exist • Behavior that differs from the right and the left • Unbounded behavior • Oscillating behavior • There are others Calculus Chapter 1

Example • If x is positive, f(x) = 1 • If x is negative, f(x) = -1 • No matter how close we get to 0, there will always be negative 1 on the left and positive 1 on the right Calculus Chapter 1

Example • As x gets closer to zero from either side, f(x) gets larger and larger • “increases without bound” • Limit does not exist Calculus Chapter 1

Example cont’d • See page 65 • You can’t always trust the picture your calculator draws • It’s wrong, but you can probably still tell there is not a limit Calculus Chapter 1

When we write We imply that the limit exists and the limit is L. If the limit of a function exists, it is unique. Note Calculus Chapter 1

Properties of Limits Calculus 1.2

Direct substitution • Works for some functions • Called continuous at c • When • This section – all limits can be evaluated this way Calculus Chapter 1

Basic limits Calculus Chapter 1

You try Calculus Chapter 1

Properties of limits • Page 71 • Can be used on all limits, even those that can’t be evaluated by direct substitution Calculus Chapter 1

Examples Calculus Chapter 1

Limits with radicals • Let n be a positive integer. The following is valid for all c if n is odd, and is valid for c > 0 if n is even. Calculus Chapter 1

Limit of a composite function • If f and g are functions such that Calculus Chapter 1

Examples • Find Calculus Chapter 1

Limits of trig functions • All can be evaluated by direct substitution • Page 74 Calculus Chapter 1

Techniques for Evaluating Limits Calculus 1.3

Indeterminate form • Direct substitution yields 0/0 • Can’t find limit directly Calculus Chapter 1

Functions that agree at all but one point • If function is undefined at point c, find another function that gives the same values for all other points, and is defined at point c. • Cancellation • Rationalization Calculus Chapter 1

Example - cancellation Calculus Chapter 1

Example - rationalization Calculus Chapter 1

The Squeeze Theorem • Page 80 Calculus Chapter 1

Two special trig limits Calculus Chapter 1

Limits with trig functions • Try to write them using one of the two special trig forms. Calculus Chapter 1

Continuity and One-Sided Limits Calculus 1.4

Continuity • A function is continuous at x = c if • There is no interruption in the graph • The graph is unbroken • There are no holes, jumps or gaps Calculus Chapter 1

Continuity • A function is continuous at c if Calculus Chapter 1

Continuity over an open interval • Continuous on open interval if continuous at each point in the interval Calculus Chapter 1

Discontinuities • When f is not continuous at c, it has a discontinuity at c • Removable discontinuity if • f can be made continuous by defining or redefining f(c) • Nonremovable Calculus Chapter 1

Discontinuities • See page 85 • Check for discontinuities where a function is undefined or in a piecewise function where the definition of the function changes Calculus Chapter 1

Limit from the right • x approaches c from values greater than c Calculus Chapter 1

Limits and Their Properties