Math 111 – Calculus I - Wilkes University



Math 111 – Calculus I.

Week Number Three Notes

Fall 2002

Limits of Functions and Methods for Computing Them

Now that we have an “intuitive notion” of what a limit is and we see why this concept is important, we will now “abstract” this notion and define the limit of a real-valued function at a value x.

Definition 3.1: Assume f is a real-valued function with domain a subset of the real line. Then, the limit as x approaches a of f(x) is L if for every positive integer ε > 0, there exists a positive integer δ > 0 such that if |x-a| < δ, then |f(x) – L| < ε. We express this utilizing the following notation.

What does the above definition mean intuitively? It says the following.

No matter HOW CLOSE we get to the number L (this is why ε in the above definition is arbitrary), we can find a distance about the x-value a (this is called δ and usually depends on ε) such that if x is less than distance δ from a (i.e. it is close to a), then f(x) is less than ε from L (i.e. it is “close” to L). NOTE: L does not have to equal f(a). IN FACT, a MAY NOT BE A MEMBER OF THE DOMAIN OF f.

The concept of limit is the concept that underlies ALL OF CALCULUS. It is ESSENTIAL that you UNDERSTAND THIS CONCEPT.

First let’s look at sketches of some functions to understand how this concept works.

Example 3.2: Consider the sketches of the following functions f(x) and g(x) below.

f(x) g(x)

Does the limit as x approaches a of f(x) exist? If so, what is it? If not, explain why not? Does the limit as x approaches a of g(x) exist? If so, what is it? If not, explain why not?

Now that we have an idea on how to estimate limits of functions graphically, we will focus on how to compute limits of functions using algebraic techniques (given an explicit definition of the function).

Example 3.3(Similar to examples 1 and 2 from pp. 102-103 of Stewart): Compute the following limits (or justify why they do not exist). Use your graphing calculator to assist in verifying the correctness of your answers.

We will learn many techniques for computing limits as this semester progresses. At this point, the following example is extremely difficult to compute using traditional algebraic techniques. In this case, technology (such as our graphing calculator) can be an invaluable tool (we learn how to explicitly evaluate and prove this limit later in the course).

Example 3.4(Sec 2.2, problem 20, parts (a) and (b) on p. 110): Consider the following function h(x).

a) Compute h(1), h(0.5), h(0.1), h(0.05), h(0.01), and h(0.005) respectively.

b) Using your result in (a), estimate the following limit.

I. Some rules for computing limits (will be stated without proof)

Theorem 3.5: Assume that f and g are functions of a real-variable x, c is a real constant and n is a positive integer. Moreover, assume that

Then,

These are the standard rules that allow us to “decompose” limits into smaller pieces that are calculable. Let’s go through two more examples to understand how these rules can be utilized.

Example 3.6: Evaluate the limits below using the limit laws. Justify each step explicitly.

Example 3.7: Assume that

Compute

using the limit laws.

The following theorem called the pinching or squeezing theorem for limits is also extremely useful. If we know two functions that have the same limit at a given value and a third function that we can “pinch” between the two functions (using inequalities) near that value, then we can calculate the third functions limit as well.

Theorem 3.8: Assume f(x) < g(x) < h(x) on an interval containing a real number a (except possibly at a). Moreover, assume that

Then,

The following example from the textbook will illustrate the use of this theorem.

Example 3.9(example 10, p. 116 of section 2.3 of Stewart): Use the pinching theorem and known limits to show that

The notion of one-sided limits

The notion of a one-sided limit is as follows. Although a function may not have a well-defined limit at a value a, it may have a limit if we approach the value from one direction (either from the left – usually called a left-hand limit or from the right – usually called a right-hand limit). I will define this concept formally now.

Definition 3.10: Assume f is a real-valued function with domain a subset of the real line. Then, the right-hand limit as x approaches a of f(x) is L if for every positive integer ε > 0, there exists a positive integer δ > 0 such that for x > a, if |x-a| < δ, then |f(x) – L| < ε. We express this utilizing the following notation.

Exercise: Analogously define the notion of a left-hand limit as x approaches a given real number.

Let’s look at one more example to get an idea of the difference between a one-sided limit versus a two-sided limit.

Example 3.11: Consider the sketches of the following functions f(x) and g(x) that we first observed in example 3.2 (refer back to page 12 of the notes). Compute left-hand and right-hand limits of f and g respectively as x approaches 1 or argue while the particular one-sided limit fails to exist.

Finally, note the following result from the textbook relating one-sided and two-sided limits of a function.

Theorem 3.12:

What does this theorem imply? Does it make intuitive sense?

Non Hand-In Homework Problems Associated with Week #3 Notes

Sections 2.2 and 2.3

Section 2.2: 1-6, 8,10,13,14,16,17,19

Section 2.3: 1-7,8,9-17, 20, 22, 23, 25, 29, 32, 36

Read Sections 2.4 through 2.5 of the textbook

EXAM #1: Friday, September 19th

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