Evaluating Limits
In this text we'll just introduce a few simple techniques for evaluating
limits and show you some examples. The more formal ways of finding limits
will be left for calculus.
A limit of a function at a certain
x-value does not depend on the value of the
function for that
x. So one technique for evaluating a limit is evaluating a
function for many
x-values very close to the desired
x. For example,
f (x) = 3x. What is
f (x)? Let's find the values of
f at some
x-values near
4.
f (3.99) = 11.97, f (3.9999) = 11.9997, f (4.01) = 12.03, andf (4.0001) = 12.0003. From this, it is safe to say that as
x approaches
4,
f (x) approaches
12. That is to say,
f (x) = 12.
The technique of evaluating a function for many values of
x near the desired
value is rather tedious. For certain functions, a much easier technique works:
direct substitution. In the problem above, we could have simply evaluated
f (4) = 12, and had our limit with one calculation. Because a limit at a given value
of
x does not depend on the value of the function at that
x-value, direct
substitution is a shortcut that does not always work. Often a function is
undefined at the desired
x-value, and in some functions, the value of
f (a)≠
f (x). So direct substitution is a technique that should be
tried with most functions (because it is so quick and easy to do) but always
double-checked. It tends to work for the limits of
polynomials and
trigonometric functions, but is less reliable for functions which are undefined
at certain values of
x.
The other simple technique for finding a limit involves direct substitution, but
requires more creativity. If direct substitution is attempted, but the function
is undefined for the given value of x, algebraic techniques for simplifying a
function may be used to findan expression of the function for which the value of
the function at the desired x is defined. Then direct substitution can be
used to find the limit. Such algebraic techniques include factoring and
rationalizing the denominator, to name a few. However a function is manipulated
so that direct substitution may work, the answer still should be checked by
either looking at the graph of the function or evaluating the function for x-
values near the desired value. Now we'll look at a few examples of limits.
What is
?
Figure 2.2: f (x) =
Direct substitution doesn't work, because
f is undefined at
x = 1. By
facotring the denominator into
(x + 1)(x - 1), though, the
(x - 1) term
cancels on the top and bottom, and we are left evaluating
. By direct substitution, the limit is
.
Consider the function
f (x) = xforx < 0, f (x) = x + 1forx≥ 0. What is
f (x), what is
f (x), and what is
f (x)?
Figure 2.3: f (x) = x for x < 0, f (x) = x + 1 for x≥ 0
The
one-sided limit from the left is
0. This we can tell both from direct
substitution and by studying the graph. Using the same techniques, we find the
one-sided limit from the right is
1. By the rules of the nonexistent limit,
f (x) does not exist, because is
f (x)≠f (x).
Consider the function
f (x) = xforallx≠3, f (x) = 2forx = 3. What is
f (x)?
Figure 2.4: f (x) = x for all x≠3, f (x) = 2 for x = 3
Direct substitution gives the limit at
2, but more careful inspection of the
graph and the values surrounding
x = 3 show that in fact the limit of
f at
x = 3 is
3. This is a prime example of how the value of a function at
x
has no impact on the limit of that function at
x.
?
By direct substitution and verification from the graph, 
.
