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Topic 7: Limits
 

 

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

  1. E. Ngezahayo & P. Icyingeneye. Advanced Mathematics for Rwanda Schools Learner's Book 4, Fountain Publishers Ltd, 2016.
 
 

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