Limit of a function properties (theorems or laws)
         Squeeze rule
         Composition rule
         Limits of functions properties use, examples
      Limits of rational functions
         Evaluating the limit of a rational function at infinity
Limit properties
Assume that  exist and that c is any constant. Then,
   
   11.   Squeeze rule
           If   f(x) < g(x) < h(x)  for all x in an open interval that contains a, except possibly at x = a
  and    
  then    
   12.  Composition rule
           If   f(x) is continuous at x = b and  
    then,    
Limits of functions properties use, examples
Limits that are commonly used are written below for easy reference,
By comparing the area of the sector of the arc x with areas of the two right triangles, the smaller triangle with legs sin x and cos x, and the bigger triangle with legs tan x and 1, shown on the right figure, we get
Since the left and the right side of the last inequality tend to the same limit 1, as x tends to 0 then, applying the squeeze rule obtained is
   
the fundamental or basic trigonometric limit. Note that the arc length x is measured in radians.
Further,
since both factors on the right tend to 1, as x tends to 0, then
   
Limits of rational functions
A rational function is the ratio of two polynomial functions
where n and m define the degree of the numerator and the denominator respectively.
Evaluating the limit of a rational function at infinity
To evaluate the limit of a rational function at infinity we divide both the numerator and the denominator of the function by the highest power of x of the denominator.
Example:  Evaluate the limit  
Solution:  
Example:  Evaluate the limit  
Solution:  
Example:  Evaluate the limit  
Solution:  
Therefore, the following three cases are possible;
1. If the degree of the numerator is greater than the degree of the denominator (n > m), then the limit of the rational function does not exist, i.e., the function diverges as x approaches infinity.
2. If the degree of the numerator is equal to the degree of the denominator (n = m), then the limit of the rational function is the ratio an/bm of the leading coefficients.
3. If the degree of the numerator is less than the degree of the denominator (n < m), then the limit of the rational function, as x tends to infinity, is zero.
The line y = c is a horizontal asymptote of the graph of f(x) if  
Thus, a rational function has a horizontal asymptote if the function values tend to a constant value c as x approaches plus or minus infinity.
Calculus contents B
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.