Lesson 1.5: Evaluating Limits — The Basics

There are several techniques for evaluating limits. Some of them require more advanced techniques in calculus, but in this lesson we will discuss the ones accessible with the methods we know so far.

The easiest way to evaluate a limit is to use the graph of the function or a CAS such as that built into the TI-89 calculator. Using the graph is straightforward. To use the TI-89, enter limit(f(x),x,a) where f(x), x, and a are from lim(x->a)f(x). To evaluate a one-sided limit, add the term ,-1 for a limit from the left or ,1 for a limit from the right before the closing parenthesis. The limit( function may be typed in manually or accessed from the F4/Calc menu.

When the use of a CAS or graph is unavailable, in some cases, it is possible to evaluate lim(x->a)f(x) by evaluating f(a). This will fail when f(x) is not equal to f(a), but this is unusual and, typically, can be easily seen from the definition of the function (often a piecewise definition). The more common reason that the evaluation of lim(x->a)f(x) by evaluating f(a) fails is that f(a) is undefined. In this case, it may be helpful to evaluate close values such as f(a+0.01) and f(a)-0.01 is. Of course, if the limit to evaluate is of the form lim(x->a+)f(x) or lim(x->a-)f(x) (a one-sided limit), be sure to evaluate only one side.

Evaluating limits at infinity such as lim(x->infinity)f(x) or lim(x->-infinity)f(x) where f(x) is a rational function is simplified by realizing the effect of the terms of the rational function as x approaches plus or minus infinity. Recall that a rational function takes the form f(x)=P(x)/Q(x) where P(x) and Q(x) are polynomials. As an example, let f(x)=(2x-1)/x. We will attempt to find lim(x->infinity)f(x). As 2x-1 and x get very large, note that the -1 term becomes irrelevant because it is very small compared to the values of 2x and x. Therefore, we can treat f1(x)=2x/x=2 (since the x cancels) for a very large x. Thus, lim(x->infinity)f(x)=2, which can be verified by examining the graph of y=f(x), having a CAS evaluate the limit, or proof with delta-epsilon as will be explained in the next two lessons.

 Graph of y=f(x)=(2x-1)/x
The graph of y=f(x)=(2x-1)/x

Finally, it is worth noting that some limits do not exist. The canonical example of this is g(x)=sin(1/x). Two views of the graph of y=g(x)this function with different windows centered around x=0 are shown. As an exercise, explore this graph further on your calculator and note that g(x) oscillates between -1 and 1 infinitely many times around x=0. For reasons that will be formalized over the next two lessons, lim(x->0)g(x) is said to not exist. Note that this is not the case of having a limit equal plus or minus infinity, which is sometimes treated as a nonexistent limit, but a function that does not approach any value—finite or infinite—at all.

 Graph of y=g(x)=sin(1/x)
Graphs of y=g(x)=sin(1/x)
Graph of y=g(x)=sin(1/x)

Suppose we want to evaluate lim(x->o)(x^2 sin(1/x)). We can look at the graph and intuit that it is equal to 0, but that’s no definitive answer. Since the expression is the product of two simpler expressions, we might be inclined to say lim(x->o)(x^2 sin(1/x))=lim(x->0)x^2 * lim(x->0)sin(1/x) by the product law, but we cannot do this because lim(x->o)sin(1/x) does not exist, as we just discussed.

There is a theorem that will help us, known as the Squeeze Theorem. It states that iff f(x) is less than or equal to g(x) is less than or equal to h(x) for any x near a except perhaps at x=a and iff lim(x->a)f(x)=L=lim(x->a)h(x), then lim(x->a)g(x)=L. So we can prove that lim(x->o)(x^2 sin(1/x))=0 by finding two functions f and h such that lim(x->o)f(x)=0 and lim(x->o)h(x)=0, x^2 sin(1/x) is greater than or equal to f(x) around x=0, and x^2 sin(1/x) is less than or equal to h(x) around x=0. The Squeeze Theorem is illustrated to the right with f(x)=x^2 and h(x)=-x^2. In our study of calculus, we will use many theorems similar to the Squeeze Theorem.

 Graph demonstrating the Squeeze Theorem
Graph of g(x)=x^2 sin(1/x)
with f(x)=x^2 and h(x)=-x^2

Some useful properties of limits, taking into account any functions f, g, and h, and a constant c and with the assumption that all the limits exist:

  • the limit of a sum is the sum of the limits (the sum law)
  • the limit of a difference is the difference of the limits (the difference law)
  • the limit of a constant times a function is the constant times the limit (the law of constant multiples)
  • the limit of a product is the product of the limits (the product law)
  • the limit of a quotient is the quotient of the limits iff the denominator is nonzero (the quotient law)
  • Iff f(x) is less than or equal to g(x) is less than or equal to h(x) for any x near a except perhaps at x=a and iff lim(x->a)f(x)=L=lim(x->a)h(x), then lim(x->a)g(x)=L (the Squeeze Theorem)

Valid XHTML 1.1!
Valid CSS!