The limit of a function
         The limit of a function examples
         Vertical, horizontal and slant (or oblique) asymptotes
         Monotone functions - increasing or decreasing in value
Vertical, horizontal and slant (or oblique) asymptotes
If a point (x, y) moves along a curve  f (x) and then at least one of its coordinates tends to infinity, while the distance between the point and a line tends to zero then, the line is called the asymptote of the curve.
Vertical asymptote
If there exists a number a such that
then the line  x = a  is the vertical asymptote.
Horizontal asymptote
If there exists a number c such that
then the line  y = c  is the horizontal asymptote.
Slant or oblique asymptote
If there exist limits
then, a line  y = mx + is the slant asymptote of the function  f (x).
Example:  Find the vertical and the horizontal
asymptote of the function
Solution:  Since,
then  x = 1  is the vertical asymptote.
Since,  
then  y = 2  is the horizontal asymptote.
Example:  Calculate asymptotes and sketch the graph of the function
Solution:  By equating the numerator with zero and solving for x we find the x-intercepts,  
x2 - x - 2 = (x + 1)(x - 2) = 0, 
x1 = -and  x2 = 2.
We calculate  f (0) to find the y-intercept,
 f (0) = 2/3.
By equating the denominator with zero and solving for x we find the vertical asymptote,
x = 3.
Let calculate following limits
to find the slant asymptote  y = mx + c.
Therefore, the line  y = x + 2 is the slant asymptote of the given function.
Example:  Find the following limits,
Solution:  The graph of the tangent function shows,
as x approaches p/2 from the left, the tangent function increases to plus infinity, while as x approaches p/2 from the right, the function decreases to minus infinity, therefore
Example:  Evaluate the limit
Solution:  
Example:  Evaluate the limits,
Solution:  The graph of the arc-tangent function shows, as x tends to minus infinity the function values approach - p/2 while, as x tends to plus infinity, the function values approach p/2.
Therefore,
and
Monotone functions - increasing or decreasing in value
1.  The function is said to be increasing if  f (x1) <  f (x2)  for all  x1 < x2
2.  The function is said to be decreasing if  f (x1) >  f (x2)  for all  x1 < x2
3.  If  f  is either increasing or decreasing then f  is said to be monotone.
Calculus contents B
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.