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Chapter 1Limit and their Properties
)(lim
xfcx
Section 1.2 Finding Limits Graphically and Numerically I. Different Approaches
A. Numerical Approach1. Construct a table of values for x and
f(x).2. Use small intervals for x to estimate
the limit that f(x) approaches.
B. Graphical Approach1. Use the table to plot points.2. Draw a graph by hand or use a
calculator.C. Analytical Approach
1. Use algebra or calculus2. Plug in the value for x that it
approaches
II. Common types of behavior associated with the nonexistence of a limit.A. f(x) approaches a different number from the right
side of c than it approaches from the left side.B. f(x) increases or decreases without bound as x
approaches c.C. f(x) oscillates between 2 fixed values as x
approaches c.III. Definition of a limit.
A. Let f be a function defined at an open interval containing c (except possibly at c) and let L be a
real number. The statementf(x) = L
Means that for each ε > 0 there exists a δ > 0 such that if 0 < |x-c| < δ , then |f(x) – L| < ε
cxlim
Section 1.3: Evaluating Limits Analytically
A. Direct Substitution
1. Factor and Cancel 2. Rationalize Numerator 3.
B. Properties of Limits
1. Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits and
a. Scalar multiple:
b. Sum or difference:
c. Product:
d. Quotient: provided K ≠ 0
e. Power: n
)()(lim cfxfcx
Lxfcx )(lim Kxgcx )(lim
bLxbfcx )(lim
KLxgxfcx )()(lim
LKxgxfcx )()(lim
K
L
xg
xfcx
)(
)(lim
Lxf ncx )(lim
C. Two Special Trigonometric Limits
1.
2.
1sin
lim 0 x
xx
0cos1
lim 0
x
xx
Section 1.4 Continuity and One-Sided Limits
A. Definition of Continuity
1. Continuity at a point : A function f is continuous at c if the following 3 conditions are met
a. f(c) is defined
b. exists
c.
2. Continuity on an open interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval
3. Removable and non-removable discontinuities.
)()(lim cfxfcx
)(lim xfcx
B. One-Sided Limits
1. Existence of a Limit: Left f be a function and let c and L be real number. The limit f(x) as x approaches c is L if and only if and
C. Continuity on a Closed Interval
1. Definition: A function f is continuous on the closed interval if it is continuous on the open interval (a,b) and and
D. Intermediate Value Theorem
1. Definition: If f is continuous on the closed interval and K is any number between f(a) and f(b), then there is at least one number c in such that f(c)=K.
Lxfcx )(limLxfcx )(lim
ba,
)()(lim afxfax
)()(lim bfxfbx
ba,
ba,
Section 1.5 – Infinite Limits
I. Infinite Limits
A. Let f be a function that is defined at every real number in some open interval containing
c (except possibly c itself).
The statement
means that for each M > 0 there exists a > 0 such that whenever
Similarly, the statement
means that for each N < 0 there exists a > 0 such that f(x) < N whenever
.To define the infinite limit from the left, replace
by c – < x < c. To define the infinite limit from the right,
replace by c < x < c + .
B. lim f(x) = does not mean that the limit exists, it tells you how the limit fails to exist by
demonstrating the unbounded behavior of f(x) as x approaches c.
)(lim xfcx
Mxf )( cx0
)(lim xfcx
cx0
cx0
cx0
II. Vertical AsymptotesA. If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f.B. 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 x c in the interval, then the graph of the function given by
has a vertical asymptote at x = c. (Vertical asymptote occurs at a number where the denominator is 0)
)(
)()(
xg
xfxh
III. 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: L > 0
L < 0
3. Quotient:
Similar properties hold for one-sided limits and for functions for which the limit of f(x) as x approaches c is
)(lim xfcx Lxgcx )(lim
)()(lim xgxfcx
)()(lim xgxfcx
)()(lim xgxfcx
0)(
)(lim xf
xgcx
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