page 1 of 2

Page 1

Page 2

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*) = 3*x*. 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 %: *f* (*x*) =

What is ?

Figure %: *f* (*x*) =

Page 1

Page 2

Take a Study Break!