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11.1. Limits and Continuity of Functions

This chapter is a brief introduction to calculus, a set of powerful mathematical tools which allow us to calculate many things which would be impossible otherwise. Once the tools are in place, calculus is not too difficult; the hard part is developing these tools. Calculus deals with * infinitesimal* quantities, i.e. quantities which can be thought of as being infinitely small. If this does not make sense to you yet, don't worry - you'll get there!

In this section we define the limit of a function. This is a necessary step toward understanding the derivative, the most basic concept in calculus.

Let f(x) be a real-valued function of a variable x with domain containing some interval (a, b), and let x_{0} be a fixed real number with a ≤ x_{0} ≤ b. Note that this might mean that x_{0} does not lie in the domain of f. This would be true, for instance, if x_{0} = a and f is defined on the interval (a, b] but not at a. We want to be able to define the * limit* of f(x) as x approaches x

_{0}. How do we do so? We say the limit of f(x) as x approaches x

_{0}is equal to y

_{0}if for all positive real numbers ε, there exists a positive real number δ such that for all numbers x in the interval (x

_{0}- δ, x

_{0}+ δ), we have |f(x) - y

_{0}| < ε. (Here we are supposed to imagine ε and δ to be small positive numbers. The Greek letters ε and δ are typically used to stand for small quantities.) In mathematical notation, this statement looks as follows:

**(11.1.1)**

We say f(x) is * continuous* at x = x

_{0}if f is defined at x

_{0}and if

**(11.1.2)**

Limits of functions are often called * two-sided limits*. There are also

*known as the*

**one-sided limits***and the*

**limit from above***. The definition of these limits are respectively as follows:*

**limit from below****(11.1.3a)****(11.1.3b)**

Note that by these definitions, a two-sided limit is both an limit from above and a limit from below.

**Example 1: **Consider the following function:

Compute the following limits if they exist, or if they do not, then say so and why. Also, determine whether f is continuous at x = 0.

- (a)
- (b)
- (c)

**Solution: **

- (a) For all positive x we have f(x) = 1. Therefore, for all ε > 0, it is certainly true that there exists a positive real number δ such that for every number x in the interval (0, δ), we have |f(x) - 1| < ε. In fact, any value of δ will work, no matter how large, for every ε > 0, no matter how small. Thus, we see that this limit from above is equal to 1.
- (b) For all negative x we have f(x) = -1. Therefore, for all ε > 0, it is certainly true that there exists a positive real number δ such that for every number x in the interval (-δ,0), we have |f(x) + 1| < ε. In fact, any value of δ will work, no matter how large, for every ε > 0, no matter how small. Thus, we see that this limit from below is equal to -1.
- (c) The two-sided limit does not exist since every interval of the form (-δ, δ) contains some values of x (namely the negative ones) for which f(x) = -1 and other values of x (namely the positive ones) for which f(x) = 1. Thus, no value of ε less than 1 will work for us.

Clearly f(x) is not continuous at x=0 since its two-sided limit is undefined there. It is continuous everywhere else, though.

**Example 2: **Consider the following function:

Compute the following limits if they exist, or if they do not, then say so and why. Also, determine whether f is continuous at x = 0.

- (a)
- (b)
- (c)

**Solution: **

- (a) We see that every interval of the form (0, δ) contains values of x as large as 1 and as small as -1, no matter how small δ is. Thus, no ε less than 1 will work for us, so this limit from above is undefined.
- (b) Similarly, we see that every interval of the form (-δ, 0) contains values of x as large as 1 and as small as -1, no matter how small δ is. Thus, no ε less than 1 will work for us, so this limit from above is undefined.
- (c) The two-sided limit clearly does not exist since neither the limit from above nor the limit from below do.

Clearly f(x) is not continuous at x=0 since its two-sided limit is undefined there. It is continuous everywhere else, though.

Limits of functions need not be finite. For instance, consider the function f(x) = 1/x, which we studied in Chapters 1 and 9. We know that f has a singularity at x=0. As x approaches zero from above, we see that f(x) = 1/x grows without limit. In this case, we say the limit of f(x) as x approaches zero from above is equal to infinity. On the other hand, as x approaches zero from below, we see that f(x) is negative and becomes more and more negative (absolute value increasing). Thus, we say that as x approaches zero from below, f(x) approaches negative infinity. Mathematically, we write these two limits as follows:

and

Since these limits are different, the two-sided limit of f(x) = 1/x at x=0 does not exist.

**Example 3:** Compute the limits from above and below of f(x) = 1/x^{2} as x approaches zero. Is the two-sided limit defined as x approaches zero?

**Solution:** f(x) = 1/x^{2} is positive for all x. As x approaches zero either from above or below, we see that f(x) approaches infinity. Thus, the limits both from above and below of f(x) as x approaches zero are equal to infinity, and the two-sided limit is as well.

Thus far, we have only considered limits of functions as x approaches some finite value. What about limits as x approaches infinity or negative infinity? These may be defined as well. Here are the definitions:

We say that the limit of a function f(x) as x approaches infinity is equal to L if for every positive number ε, there exists a positive number N such that for all x greater than N, we have |f(x) - L| < ε. This is written mathematically as follows:

The limit of a function as the argument approaches -∞ is defined similarly. We say that the limit of a function f(x) as x approaches -∞ is equal to L if for every positive number ε, there exists a positive number N such that for all x less than -N, we have |f(x) - L| < ε. This is written mathematically as follows:

If a function f(x) has a finite limit L as x approaches infinity, then the function approaches the line y = L as x gets large. Thus, we see that this line is an asymptote to f. Similarly, if the limit of f(x) as x approaches negative infinity is equal to L, then the function approaches the line y = L as x gets small (negative and large absolute value), so the line y = L is an asymptote of f in this case as well.

Limits of rational functions as x approaches infinity or negative infinity are easy to compute. The trick to computing these limits is to only consider the leading terms of the both the numerator and denominator.

**Example 3: **Compute the limits of the following rational functions as x approaches infinity and negative infinity.

- (a) f(x) = 1 / (x - 3)
- (b) f(x) = 1 / (1 - x
^{2}) - (c) f(x) = (x
^{2}+ 1) / (x^{2}- 1) - (d) f(x) = (x
^{2}- 4) / (3x^{2}+ 8x - 3) - (e) f(x) = (6x
^{3}- 5x + 4) / (2x^{3}+ 7x^{2}+ 6) - (f) f(x) = (x
^{2}+ x + 1) / (10 - 3x) - (g) f(x) = (x
^{3}+ x + 1) / (10 - 3x) - (h) f(x) = (1 - x
^{3}) / (3 - x)

**Solution: **

- (a) As |x| approaches infinity, f(x) behaves like 1/x, which approaches zero as x approaches infinity or negative infinity. Thus, both limits are equal to 0.
- (b) As |x| approaches infinity, f(x) behaves like -1/x
^{2}, which approaches zero as x approaches infinity or negative infinity. Thus, both limits are equal to 0. - (c) As |x| approaches infinity, f(x) behaves like x
^{2}/ x^{2}= 1. Thus, both limits are equal to 1. - (d) As |x| approaches infinity, f(x) behaves like x
^{2}/ 3x^{2}= 1/3. Thus, both limits are equal to 1/3. - (e) As |x| approaches infinity, f(x) behaves like 6x
^{3}/ 2x^{3}= 3. Thus, both limits are equal to 3. - (f) As |x| approaches infinity, f(x) behaves like x
^{2}/ -3x^{}= -x/3, which approaches negative infinity as x approaches infinity and approaches infinity as x approaches negative infinity. - (g) As |x| approaches infinity, f(x) behaves like x
^{3}/ -3x^{}= -x^{2}/3, which approaches negative infinity as x approaches infinity or negative infinity. - (h) As |x| approaches infinity, f(x) behaves like -x
^{3}/ -x^{}= x^{2}, which approaches infinity as x approaches infinity or negative infinity.

To close this section, we state without proof some important properties of limits, which we will find useful.

**(11.1.4)****(11.1.5)****(11.1.6)****(11.1.7)**

As usual, we must be careful with these properties, because some of the above limits may not exist. Strictly speaking, the above properties are only valid if all limits shown exist.