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Ph.D. Math, UC Berkeley

 

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>> 11.2. Derivatives

 

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 x0 be a fixed real number with a ≤ x0 ≤ b. Note that this might mean that x0 does not lie in the domain of f. This would be true, for instance, if x0 = 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 x0. How do we do so? We say the limit of f(x) as x approaches x0 is equal to y0 if for all positive real numbers ε, there exists a positive real number δ such that for all numbers x in the interval (x0 - δ, x0 + δ), we have |f(x) - y0| < 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)two-sided function limit definition

We say f(x) is continuous at x = x0 if f is defined at x0 and if

  • (11.1.2)continuity definition

Limits of functions are often called two-sided limits. There are also one-sided limits known as the limit from above and the limit from below . The definition of these limits are respectively as follows:

  • (11.1.3a)limit from above definition
  • (11.1.3b)limit from below definition

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:

example 1 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) limit from above
  • (b) limit from below
  • (c)limit

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:

example 2 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) limit from above
  • (b) limit from below
  • (c)limit

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:

limit of 1/x as x approaches zero from above and limit of 1/x as x approaches zero from below

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/x2 as x approaches zero. Is the two-sided limit defined as x approaches zero?

Solution: f(x) = 1/x2 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:

function limit as x approaches infinity

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:

function limit as x approaches negative infinity

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 - x2)
  • (c) f(x) = (x2 + 1) / (x2 - 1)
  • (d) f(x) = (x2 - 4) / (3x2 + 8x - 3)
  • (e) f(x) = (6x3 - 5x + 4) / (2x3 + 7x2 + 6)
  • (f) f(x) = (x2 + x + 1) / (10 - 3x)
  • (g) f(x) = (x3 + x + 1) / (10 - 3x)
  • (h) f(x) = (1 - x3) / (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/x2, 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 x2 / x2 = 1. Thus, both limits are equal to 1.
  • (d) As |x| approaches infinity, f(x) behaves like x2 / 3x2 = 1/3. Thus, both limits are equal to 1/3.
  • (e) As |x| approaches infinity, f(x) behaves like 6x3 / 2x3 = 3. Thus, both limits are equal to 3.
  • (f) As |x| approaches infinity, f(x) behaves like x2 / -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 x3 / -3x = -x2/3, which approaches negative infinity as x approaches infinity or negative infinity.
  • (h) As |x| approaches infinity, f(x) behaves like -x3 / -x = x2, 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) additive property of limits
  • (11.1.5) limit of difference
  • (11.1.6) multiplicative property of limits
  • (11.1.7) quotient property of limits

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.

 

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