discussions in chapter 13 on limits were restricted to the usage of the
natural numbers. We are now ready to discuss limits in a broader
sense. We can now apply the concept of limits to all real
numbers. We can also
extend the idea of a limit to include studying a limit as it approaches
finite number instead of always approaching one of the two
infinities. We will take a look at some examples of limits and
also discuss some techniques for evaluating limits. We will then
relate the concept of limits to continuous/discontinuous
functions! Hang on to your hat!
limits on infinity.
each of the following limits:
Using your calculator, you will
get the following table:
As you study the y values which
represent the limit, you can see that the numbers are getting
arbitrarily big. Therefore, the limit is positive infinity.
calculator, you will again get the following table:
Again, looking at the y values, leads
you to conclude that as you take larger and larger negative numbers,
the limit approaches negative infinity.
limit as x approaches some real number c!
1) , means
the limit of f(x) as x approaches c from the right!
2) , means the
limit of f(x) as x approaches c from the left!
the graph to illustrate the above two limits!
the graph to find each: and . To find the first limit, follow the
graph from the far left until that path comes to an end. What y
value does it approach? From the graph, it approaches 3!
Now for the second limit. Follow the
graph from the far right until the path comes to an end. This one
approaches the y value -2! Notice, that neither limit approached
the actual value of f(2) which is 1! Also note that the limit as
x approaches 2 does not exist! Why? Both limits do not approach
the same value!!
Remember the 3 possibilities from chapter 13?
the lim f(x) and lim g(x) both exist and lim g(x) 0, then:
theorem helps find many limits but it must be in the correct
that both limits must exist and lim g(x) 0. Then to find the limit
is easy. It's the limit of the numerator over the limit of the
not many limits can be solved using the above formula. Most
fit neatly into the correct form. So you might find the following
in determining limits.
If possible, use the quotient theorem. Check this first.
If lim f(x) = 0 and lim g(x) = 0, try one of the following:
Factor f(x) and g(x) and reduce to lowest terms.
If f(x) or g(x) involves a square root, multiply both the numerator and
the denominator by the conjugate of the square root expression.
If lim f(x) 0 and lim g(x) = 0 then one of the following is true:
If x approaches infinity or negative infinity, divide the numerator and
the denominator by the highest power of x in the denominator. (This sounds familiar!)
If all else fails, get out the calculator and crunch the numbers.
If x approaches infinity or negative infinity use very large or very
small values and watch the pattern develop. If the limit
approaches a number c, then
use numbers close to c on both sides of c.
1) Because the
problem has a root in it we will try 2b. 2) Because the lim
of the num and den = 0, try 2a. 3) Because the limit of
the num and dem both exist, you can use 1. 4) Because the lim
f(x) is not zero, and the lim g(x) = 0, it
must be choice 3.
the chart you can see that as the numbers get closer to 0 from the
right, the limit is infinity, but as you approach from the left, the
limit is negative infinity. Therefore, the limit does not exist! 5)
Because the limit goes
toward infinity it's #4.
Continuous vs. Discontinuous Functions
a continuous function is a function you can draw without lifting your
pencil. There are no breaks in the graph or holes in the
graph. But to say
that a function that is defined for all values of x makes it a
function is not accurate. There are many functions that have the
numbers as their domain and the function is not continuous. A
formal definition is as follows:
Formal definition of continuous:
A function is continuous at a real number c if:
provided these 3 conditions are met:
1) must exist.
one must = statement two!
following graphs give examples of discontinuous functions: