# 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*) = 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 ?

*f*(

*x*) =

By direct substitution and verification from the graph, = – .

What is ?

*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 .

*f*(

*x*) =

*xforx*< 0,

*f*(

*x*) =

*x*+ 1

*forx*≥ 0 . What is

*f*(

*x*) , what is

*f*(

*x*) , and what is

*f*(

*x*) ?

*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*) = 2*forx* = 3 . What is *f* (*x*) ?

*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* = 3is 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* .