Part 7
Infinite Limits
Definition of Infinite Limits (無窮極限)
Let f be a function that is defined at every real number in some open interval containing
point c (except possibly at c itself). The statement
l i mf x( )
x c
means that for each M  0 , there exists a   0 such that
f ( x)  M , whenever 0  x  c   .
Continuity (連續)
A function f is continuous at c if the following three conditions are met.
① f (c ) is defined
② lim f ( x ) exists
x c
③ lim f ( x)  f (c)
x c
    0,   0,  0  x  c    f ( x)  f (c)   
The Intermediate Value Theorem (中間值定理)
If f is continuous on the closed interval [a, b] and k is any real number between
f ( a ) and f (b ) , then there is at least one number c in [a, b] such that f (c )  k .
Asymptote
1. Definition of Vertical Asymptote (垂直漸近線)
If f ( x) approaches infinity (or negative infinity) as x approaches c from the right or
1
the left, then the line x  c is a vertical asymptote of the graph of f .
(i.e., lim f ( x)   or lim f ( x)   )
x c
x c
2. Definition of Horizontal Asymptote (水平漸近線)
The line y  b is a horizontal asymptote of the graph of f if
l i mf x( ) b or lim f ( x)  b
x  
x 
3. Definition of Slant Asymptote (斜漸近線)
The line y  l ( x)  mx  b , m  0 , is a slant asymptote of the graph of f if
l i m f x( ) l x( ) or0 lim  f ( x)  l ( x)   0
x  
x 
2