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Notes Chapter 1(Limits)
Existence of Limit:
Common Types of Behavior Associated with Nonexistence of a Limit:
1. approaches a different number from the right side of c then the left side of c. 2.. 3.
Definition of a Limit: Let be a function defined on an open interval containing c (except
possibly at c) and let L be a real number. The statement means that for each
ε > 0 there exists a ϭ > 0 such that if 0 < | x – c | < ϭ, then | < ε.
Basic Limits: Let b and c be real numbers and let n be a positive interger.
**1.
3.**
Limit of a Composite Function: If and g are functions such that and
, then
Limits of Trigonometric Functions: Let c be a real number in the domain of the given trigonometric function.
Etc… for all other trig functions.
Functions That Agree at All But One Point: Let c be a real number and let for all
x c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of also exists and.
Squeeze Theorem: If h(x) g(x) for all x in an open interval containing c, except
possibly at c itself, and if then exists and is equal to L.
Two Special Trigonometric Limits:
1.
2.
Definition of Continuity: Continuity at a point: a function is continuous at c if the following
three conditions are met.
1. is defined.
-Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continous at each point in the interval. A function that is continous on the entire real line (- ) is everywhere continuous.
The Existence of a Limit: Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if.
Definition of Continuity on a Closed Interval: A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and and
. The function f is continuous from the right at a and continuous from the left at b.
Properties of Continuity: If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c.
1. Scalar multiple b f 2. Sum and difference f g 3. Product f g 4. Quotient: , if g(c) 0
Continuity of a Composite Function: If g is continuous at c and f is continuous at g(c), then the composite function given by ( f g)(x) = f (g(x)) is continuous at c.
Intermediate Value Theorem: If f is continuous on the closed interval [a,b] and k is any number between f (a) and f (b), then there is at least one number c in [a,b] such that f (c) = k.
Definition of Infinite Limits: Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement means that for each M > 0 there exists a ϭ > 0 such that f (x) > M whenever 0 < | x-c | < ϭ. Similarly, the statement means that for each N < 0 there exists a ϭ > 0 such that f
(x) < N whenever 0 < |x-c| < ϭ. To define the infinite limit from the left, replace 0 < |x-c| < ϭ by c- ϭ < x <c. To define the infinite limit from the right, replace 0 < |x-c| < ϭ by c < x < c + ϭ.
Vertical Asymptotes: If f(x) approaches infinity (or negative infinity) as x approaches c from the right or left, then the line x=c is a vertical asymptote of the graph of f. Let f and g be continuous on an open interval containing c. If f (c) 0, g(c) = 0, and there exists an open interval containing c such that g(x)0 for all xc in the interval, then the graph of the function
given by h(x) = has a vertical asymptote at x = c.
Properties of Infinite Limits: Let c and L be real numbers and let f and g be functions such that and.
1. Sum or Difference: 2. Product: (Depending on if L>0 or L <0) 3. Quotient: = 0
