1. DEFINITIONS
Limits are used to describe how a function behaves as the independent variable moves towards a certain values. Frequently when studying function , we find ourselves interested in the function’s behavior near a particular point , but not at .
Let us explore numerically how the function behaves near x = 3:
Note that is defined for all real numbers x except for x = 3. For any we can simplify the expression by factoring the numerator and canceling common factors:
Even though is not defined, it is clear that we can make the value of as close as we want to 6 by choosing x close enough to 3. Therefore, we say that approaches arbitrarily close to 6 as x approaches 3, or, more simply, approaches the limit 6 as x approaches 3.
We write this as
Mathematical denotation
if and only if or in terms of intervals we can write
Example
Sow that
Solution
Given , we must find such that if .
Now,
But . Hence given , it is sufficient to choose .
According to the definition
2. COMPUTATIONAL TECHNICS
2.1. Limit of algebraic functions
Before we start working with algebraic functions, let look on some important results:
Notice: An indeterminate form is a certain type of expression with a limit that is not evident by inspection.
There are several types of indeterminate forms such as
Limit of a constant function
Since a constant function has the same value every where, it follows that at each point .
Example
Limit of a polynomial function
Limit of a polynomial as x approaches a is found by substituting this value of a in the given function.
Example
Example
Limit of a rational function
Remember that a rational function has the form . To find limit of this function, substitute for the unknown and check the result.
Example
Example
Example
Example
Limit of Irational functions
When we are computing the limits of irrational functions, in case of indeterminate form, we need to know the conjugate of the irrational expression in that function. We may need to find the domain of the given function.
Example
Example
Example
But,
2.2. One sided limits
Example
Find
Solution
We see that , then does not exist.
Example
Find
Solution
2.3. Finding limits graphically
Limits can be found graphically.
Example
Consider the following pragh
Find
Solution
From the graph,
2.4. Rules for calculating limits
Let . Then,
Theorem: The Sandwich theorem (or Squeeze theorem or Pinching theorem)
Example
Find
Solution
2.5. Limits of trigonometric functions
Before starting, let find
Consider the following tables:
From the above tables, we see that the values of approaches 1 as x is approaching 0 from left and from right. Then,
Example
2.6. Limits of logarithmic functions
3. CONTINUITY AND DISCONTINUITY OF A FUNCTION
A function is said to be continuous at point k if the following conditions are satisfied
If one or more conditions in this definition fail to hold, then f is said to be discontinuous at point k, and k is called a point of discontinuity of f.
Example
The function is discontinuous at 3 because is not defined.
4. ASYMPTOTES OF A FUNCTION
A straight line that is closely approached to the curve so that the perpendicular distance between them decreases to zero is said to be an asymptote of that curve. We need to determine the domain of definition and evaluate the limits at the boundaries of that domain in order to find the asymptotes. There is a vertical asymptote, a horizontal asymptote and an oblique asymptote.
 If then is a vertical asymptote.
 If then is a horizontal asymptote
 The line is an oblique asymptote
Example
Find relative asymptotes for the following functions
1)
2)
Solution
1)
Since there is HA as , there is no oblique asymptote.
2)
As there is no horizontal asymptote on the right side, let us check if there is oblique asymptote.
5. APPLICATIONS
References
 E. Ngezahayo & P. Icyingeneye. Advanced Mathematics for Rwanda Schools Learner's Book 4, Fountain Publishers Ltd, 2016.
