Limits and Continuity

Informal Definition of the Limit1

Informal Definition
If the values of a function f(x) approach the value L as x approaches c, we say that f(x) has the limit L as x approaches c, and we write

In simpler terms, as x approaches c, the value of f(x) approaches L.

Example 1
As the following graph and table suggest, (2x + 1) = 5.


Example 2
If f is the identity function f(x) = x, then for any value of c:
f(x) = (x) = c


Example 3
If f is the constant function f(x) = k (the function whose outputs have the constant value k), then for any
value of c:
f(x) = (k) = k


Example 4
If c is any number, 2x2 = 2c2.


Example 5




Properties of Limits1

1. Sum Rule:[f1(x) + f2(x)]  = f1(x)   + f2(x)
2. Difference Rule:[f1(x) - f2(x)]  = f1(x)   - f2(x)
3. Product Rule:[f1(x) f2(x)]  = f1(x)    f2(x)
4. Constant Multiple Rule:[k f2(x)]  = k f2(x)
5. Quotient Rule:

In words, these properties say:
  1. The limit of the sum of two functions is the sum of their limits.
  2. The limit of the difference of two functions is the difference of their limits.
  3. The limit of a product of two functions is the product of their limits.
  4. The limit of a constant times a function is the constant times the limit of the function.
  5. The limit of a quotient of two functions is the quotient of their limits, provided the denominator does not tend to zero.

Example 6
We know from Examples 2 and 3 that x = c and k = k. The various properties of limits now let us combine these results to calculate other limits:

There is one last property of limits that we must consider. The limit of a function f(x) as x approaches c never depends on what happens when x = c. The limit, if it exists at all, is entirely determined by the values that f has when x is not equal to c. You can see this in the next example.

Example 7
If
then f(x) = (x) = 2 while f(2) = 3.

As always, the limit is determined by the function's approach behavior, not by what happens at x = 2 (Figure 1).
Figure 1: The graph of .
Notice that f(x) approaches 2 as x approaches 2 even though f(2) itself is 3.


Solving by Substitution1

Limits of polynomials can always be found by substitution.
If f(x) = anxn + an-1xn-1 + . . . + a0 is any polynomial function, then
f(x) = f(c) = ancn + an-1cn-1 + . . . + a0


Limits of (many but not all) rational functions can be found by substitution.
If f(x) and g(x) are polynomials, then


Example 8

a) x2(2 - x) = (2x2 - x3) = 2(3)2 - (3)3 = 18 - 27 = -9.

b) Same limit, found another way:
  x2(2 - x) = x2(2 - x) = (3)2(2 - 3) = 9(-1) = -9.


Example 9

.


In Example 9, we can use Equation (2) to find the limit of f(x)/g(x) because the value of the denominator, g(x) = x + 2, is not equal to zero when x = 2.

Solving by Factoring2


In many cases, direct substitution will produce a meaningless fractional form 0/0. We call such an expression an indeterminate form, since we cannot (from the form alone) determine the limit. When you are trying to evaluate a limit and encounter this form, remember that you must change the fraction so that the new denominator does not have zero as its limit. One way to do this is to cancel like factors, as shown in the next two examples.

Example 10

Find   


Solution: The denominator is 0 when x = 2, so we cannot calculate the limit by substitution. However, if we factor the numerator and denominator we find that

Is it really all right to cancel terms like this? Yes, because 2 is not in the domain of the original function
(x3 - 8)/(x2 - 4), so we're not dividing by zero in disguise when we cancel the two (x-2)s.
With the (x - 2) out of the way, we can now find the limit by substitution:
(The algebra we just did)
(Equation (2) now applies)
 

Example 11

Substitution will not give the limit because x - 5 = 0 when x = 5.
We factor the numerator to see if (x - 5) is a factor. It is. We cancel the (x-5)'s, leaving. . .
. . .an equivalent form whose limit we can now find by substitution.

Solving by Rationalization2

A second way to find the limit of a function for which direct substitution yields the indeterminate form 0/0 is to use a rationalization technique. This technique can be used to rationalize either the numerator or the denominator. For instance, in the next example we begin by rationalizing the numerator -- then we cancel common factors in the rationalized form.

Example 12
Find the folowing limit:

Solution: By direct substitution we get the indeterminate form 0/0:

In this case, we change the form of the fraction by rationalizing (eliminating the radical in) the numerator:

Therefore, by substitution, we have


Limits From Different Sides1

Sometimes the values of a function f(x) tend to different limits as x approaches a number c from different sides. When this happens, we call the limit of f as x approaches c from the right the right-hand limit of f at c, and the limit as x approaches c from the left the left-hand limit" of f at c.

The notation for the right-hand limit is
f(x)      ("the limit of f as x approaches c from the right")
The (+) is there to say that x approaches c through values above c on the number line.
The notation for the left-hand limit is
f(x)      ("the limit of f as x approaches c from the left")
The (-) is there to say that x approaches c through values below c on the number line.

Example 13
Figure 2: At each integer, the greatest integer function y = has different right-hand and left-hand limits.
The greatest-integer function f(x) = has different right-hand and left-hand limits at each integer. As we can see in Figure 2,
The limit of as x approaches an integer n from above is n, while the limit as x approaches n from below is n - 1.



We sometimes call the two-sided limit of f at c to distinguish it from the one-sided right-hand and left-hand limits of f at c. If the two one-sided right-hand and left-hand limits of f(x) exist at c and are equal, their common value is the two-sided limit limit of f at c. Conversely, if the two-sided limit of f at c exists, the two one-sided limits exist and have the same value as the two-sided limit.

Relationship between One-sided and Two-sided Limits
A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In symbols,
=  L    <=>  =  L   and =  L    (3)


Example 14
All the following statements about the function y = f(x) graphed in Figure 3 are true.
Figure 3: Example 13 discusses the limit properties of the function y = f(x) graphed here.


At every other point c between 0 and 4, f(x) has a limit as x approaches c.

Example 15
The greatest-integer function f(x) = has no limit as x approaches 3. As we saw in Example 13, while . Since the right-hand and left-hand limits of f at 3 are not equal, the function has no single limiting value as x approaches 3.


Example 16
Show that .
Solution: We prove that by showing that the right-hand and left-hand limits are both 0:

In the examples we have seen so far, the functions that failed to have limits at various points did so because the right-hand and left-hand limits at the points were not equal. The function in the next example fails to have a limit because neither the right-hand limit nor the left-hand limit exists at all.

Example 17
Show that the function y = sin(1/x) has no limit as x approaches 0 from either side (Figure 4).

Figure 4: The function f = sin(1/x) has neither a right-hand nor a left-hand limit as x approaches 0.
Solution: As x approaches 0, its reciprocal, 1/x, becomes infinite and the values of sin(1/x) cycle repeatedly from -1 to 1. Thus there is no single number L that the function's values all get close to as x approaches zero. This is true even if we restrict x to positive values or to negative values. The function has neither a right-hand limit nor a let-hand limit as x approaches 0.

Continuity2

Figure 5: Three points of discontinuity
In mathematics the term continuous has much the same meaning as it does in our everyday usage. To say that a function is continuous at x = c means that there is no interruption in the graph of f at c. That is, its graph is unbroken at c and there are no holes jumps, or gaps. For example, Figure 5 identifies three values of x at which the graph of f is not continuous. At all other points of the interval (a, b), the graph of f is uninterrupted, and we say it is continuous at such points. Thus, it appears that the continuity of a function at x = c can be destroyed by any one of the following conditions:
  1. The function is not defined at x = c.
  2. The limit of f(x) does not exist at x = c.
  3. The limit of f(x) exists at x = c, but is not equal to f(c).

Definition of continuity
A function f is said to be continuous at c if the following three conditions are met:
1. f(c) is defined.   2.f(x) exists.   3. f(x) = f(c)
A function is said to be continuous on an interval (a, b) if it is continuous at each point in the interval.

Figure 6
A function f is said to be discontinuous at c if f is defined on an open interval containing c (except possibly at c) and f is not continuous at c. Discontinuities fall into two categories: removable and nonremovable. A discontinuity at x = c is called removable if f can be made continuous by redefining f at x = c. For example, in Figure 6 the function f has two removalbe discontinuities and one nonremovable discontinuity.

Example 18
Determine whether the following functions are continuous on the given interval:
Solution: The graphs of these three functions are shown in Figure 7.
  1. Since f is a rational function whose denominator is not zero in the interval (0,1), we can conclude that f is continuous on (0,1).
  2. Since f is undefined at x = 1, we conclude that it is discontinuous at x = 1. Since the denominator is not zero at any other point on the interval, we can conclude that f is continuous at all other points in the interval (0,2).
  3. Since polynomial functions are defined over the entire real line, we can see that f is continuous on (-,).
Figure 7

Each of the intervals in Example 18 is open. To discuss continuity on a closed interval, we use the concept of one-sided limits.

Definition of Continuity on a Closed Interval
If f is defined on a closed interval [a, b], continuous on (a, b), and

the f is said to be(x) continuous on [a, b].

Example 19
Discuss the continuity of
Figure 8
Solution: We know that the polynomial functions given by 5 - x and x2 - 1 are continuous for all real x. Thus, to conclude that g is continuous on the entire interval [-1,3], we need only check the behavior of g when x = 2. By taking the one-sided limits when x = 2, we see that

and

Since these two limits are equal, we can conclude that
g(x) = g(2) = 3

Thus, g is continuous at x = 2, and consequently it is continuous on the entire interval [-1,3]. The graph of g is shown in Figure 8.

Properties of Continuous Functions
If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c.
  1. Scalar multiple: bf
  2. Sum and difference: fg
  3. Product: fg
  4. Quotient: f/g, if g(c) is not = 0

Now we can look at the continuity of composite function, such as f(x) = (x2 + 1)1/2.
Continuity of a Composite Function
If g is continuous at c and f is continuous at g(c), then the composite function given by fg(x) = f(g(x)) is continuous at c

Example 20
Find the intervals for which the three functions shown in Figure 9 are continuous.
Solution:
  1. The function f(x) = (1 - x2)1/2 is continuous on the closed interval [-1,1].
  2. The function f(x) = 1/(1 - x2)1/2 is continuous on the open interval (-1,1). (Note that f is undefined for all x such that x>=1.)
  3. At x = 1, the limits from the right and left are zero. Thus, the function f(x) = x2 - 1 is continuous on the entire real line, that is, the interval (-,).

Figure 10
The Intermediate Value Theorem states that for a continuous function f, if x takes on all values between a and b, then f(x) must take on all values between f(a) and f(b). As a practical example of this theorem, consider a person's height. Supposed that a boy is 5 feet tall on his thirteenth birthday and 5 feet 7 inches tall on his fourteenth birthday. Then, for any height h between 5 feet and 5 feet 7 inches, there must have been a time t when his height was exactly h. This seems reasonable, since we believe that normal human growth is continuous and a person's height could not abruptly change from one value to another.

Intermediate Value Theorem
If f is continuous on [a,b] and k is any number between f(a) and f(b), then there is at least one number c between a and b such that f(c) = k.

Figure 11
The Intermediate Value Theorem states that if the domain of a function is a closed interval and the function is continuous, then there are no holes or gaps in the graph of the function. Discontinuous functions, however, may not possess the intermediate value property, as a comparison of Figures 10 and 11 indicates.
You may be familiar with one application of the Intermediate Value Theorem in algebra. That is, if we are given a polynomial function f and there are two numbers a and b such that f(a) is negative and f(b) is positive, then the function must have a zero (its graph must cross the x-axis) between a and b.

Example 21

Figure 12
Use the Intermediate Value Theorem to show that the polynomial function f(x) = x3 + 2x - 1 has a zero in the interval [0,1].
Solution: Since f(0)=03+2(0)-1=-1 and f(1)=13+2(1)-1=2
we can apply the Intermediate Value Theorem to conclude that there must be some c in (0,1) such that f(c)=0, as shown in Figure 12.

Infinite Limits1

Limits as x Approaches


Figure 13: The graph of y = 1/x
The function f(x) = 1/x is defined for all real numbers except x = 0. As Figure 13 suggests,
  1. 1/x is small and positive when x is large and positive.
  2. 1/x is large and positive when x is small and positive.
  3. 1/x is large and negative when x is small and negative.
  4. 1/x is small and negative when x is large and negative.
We summarize these facts by saying:
  1. As x approaches , 1/x approaches 0.
  2. As x approaches 0 from the right, 1/x approaches .
  3. As x approaches 0 from the left, 1/x approaches -.
  4. As x approaches -, 1/x approaches 0.
The symbol , infinity, does not represent any real number. We cannot use in arithmetic in the usual way, but it is convenient to be able to say things like "the limit of 1/x as x approaches infinity is 0."

Calculation Rules for Functions with Finite Limits as x Approaches
First we'll find the limits of two "basic" functions as x approaches and -.
Example 22
If f(x) = 3 is the constant function whose outputs have the constant value k, then
f(x) = (3) = 3
f(x) = (3) = 3


Example 23
1/x = 0

As x approaches , the graph of 1/x gets arbitrarily close to the x-axis, in the following sense: No matter how small a positive number you name, the value of 1/x eventually gets smaller.

Believe it or not, we now have enough specific information to calculate the limits of a wide variety of rational functions as x approaches . To do so, we simply use the same properties as we did for limits as x approaches c.

Example 24


Limits of Rational Functinos as x Approaches
To find the limit of a rational function as x approaches (when the limit exists), we divide the numerator and denominator by the highest power of x in the denominator.

Example 25


Example 26
We could have multiplied he original fractions together first, to express the product as a single rational function, but it would have taken longer to get the answer that way.


The rule for finding limits of rational functions as x approaches is this: If the numerator and denominator have the same degree, the limit is the ratio of the leading coefficients. If the degree of the numerator is less than the degree of the denominator, the limit is zero. If the degree of the numerator is greater than the degree of the denominator, the limit is infinite, as we shall see in a moment.
Lim f(x) = or lim f(x) = -

As suggested by the behavior of 1/x as x approaches 0, we sometimes want to say such things as
f(x) =
f(x) =
f(x) =
f(x) =
f(x) =

In every instance, we mean that the value of f(x) eventually exceeds any positive real number B. That is, for any real number B no matter how large, the values eventually satisfy the condition
f(x) > B.

Similarly, we write
f(x) = -, f(x) = -, f(x) = -

to say that no matter how large the negative number -B may be numerically, the values of f eventually satisfy the condition
f(x) < -B


Example 27


Example 28


The procedure in Example 28 is typical. To find the limit as x approaches of a rational function in which the degree of the numerator exceeds the degree of the denominator, divide by the highest power of x in the denominator. The limit of the new denominator will then be finite, and the limit of the new numerator will be infinite. The limit of the ratio can be either + or -, depending on the signs assumed by the numerator and denominator as x becomes numerically large.
Summary for Rational Functions
  1. is 0 if deg(f) < deg(g)
  2. is finite if deg(f) = deg(g)
  3. is infinite if deg(f) > deg(g)


Sometimes a change of variable can turn an unfamiliar expression into one whose limit we know how to find. Here is an example.

Example 29