 The limit of a function
The limit of a function examples
Monotone functions - increasing or decreasing in value
The limit of a function examples
 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 = -1  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.   Functions contents E 