• In this section, the limit is described, examples of limits are given, as well as rules governing limits of functions that are added, subtracted, multiplied, divided, and raised to a power.
• The limit is used in the estimation of infinite sums and the definition of the derivative in calculus.
• In general, the limit is used to describe closeness of a function to a value when the exact value cannot be identified.
• If the Limx→af(x) = L, then it is said that the function f(x) gets close to and may equal some number L as x approaches and gets close to some number a.
• For example, consider the function:
f(x) = x2 + 2
What will the value of f(x) be close to when x is close to 1?
Limx→1f(x) = Limx→1(x2 + 2)
For x = 0.9, f(0.9) = (0.9)2 + 2 = 2.8
For x = 1.1, f(1.1) = (1.1)2 + 2 = 3.2
For x = 0.99, f(0.99) = (0.99)2 + 2 = 2.98
For x = 1.01, f(1.01) = (1.01)2 + 2 = 3.02
As x gets closer to 1, Limx→1, then x2 gets closer to 1 and f(x) gets closer to 3.
Limx→1f(x) = Limx→1(x2 + 2) = 3
• Example: Consider the limit:
The limit cannot be found by substituting into the equation. For x = 2:
This form of the equation does not enable us to determine if a limit exists.
Instead, by factoring, it is possible to determine if the limit exists. (See Section 5.7 in Chapter 5, “Polynomials,” in the second book of the Master Math series, Algebra for a review of factoring polynomials.)
Taking the limit:
Limx→2(x + 2) = 4
Therefore, as x gets close to 2, x + 2 gets close to 4.
• Example: Consider the limit as y gets close to zero for the function:
f(y) = 1/y
Limy→0(1/y)
Because 1/0 is undefined, what will happen to f(y) as y gets close to 0?
If y = 0.1, f(y) = 1/0.1 = 10
If y = 0.01, f(y) = 1/0.01 = 100
If y = 0.001, f(y) = 1/0.001 = 1,000
If y = –0.01, f(y) = 1/–0.01 = –100
If y = –0.001, f(y) = 1/–0.001 = –1,000
As y gets close to 0, 1/y does not get closer to any number. The magnitude of 1/y actually increases.
Therefore, Limy→0(1/y) approaches ∞, and the limit does not exist.
• Example: Consider the limit as x gets close to infinity for the function:
f(x) = 1/x
Limx→∞(1/x)
As x gets very large what happens to 1/x?
If x = 1,000,000,
then f(x) = 1/1,000,000 = 0.000001 = 10–6
If x = 1,000,000,000,
then f(x) = 1/1,000,000,000 = 0.000000001 = 10–9
If x = 1020,
then f(x) = 1/1020 = 0.00000000000000000001 = 10–20
Therefore, as x becomes large and approaches ± ∞, 1/x becomes very small and approaches zero.
Limx→∞(1/x) = 0
• If the limits of the functions f(x) and g(x) exist, then the rules governing limits of functions that are added, subtracted, multiplied, divided, and raised to a power are:
Limx→a(f(x) + g(x)) = Limx→af(x) + Limx→ag(x)
Limx→a(f(x) – g(x)) = Limx→af(x) – Limx→ag(x)
Limx→a(f(x) × g(x)) = Limx→af(x) × Limx→ag(x)
Limx→a(f(x) ÷ g(x)) = Limx→af(x) ÷ Limx→ag(x)
provided that Limx→ag(x) ≠ 0
Limx→a(f(x))y = (Limx→af(x))y
• The limit is used to determine if functions are continuous functions or discontinuous functions. Whether or not the graph of a given function is a smooth and continuous curve or line, or whether there are breaks or holes present can be determined using the limit.
• A function is considered continuous at x = a, if Limx→a exists, Limx→a = f(a) and f(a) is defined.
• Whether a function is continuous over the part of the curve near a chosen point, can be determined using the following method. To evaluate whether a function is continuous:
Choose a pair of numbers in the domain set and corresponding range set, relating to the X-axis and Y-axis respectively of the graph.
Take the limit as x gets close to the chosen value in the domain set by choosing x values that are close to x and substituting them into the function.
If the function is continuous, the resulting f(x) values in the range set will be close to the chosen f(x) value.
• For example, consider the function:
f(x) = 1 + x
To graph, choose values for x and calculate the corresponding f(x) values.
Values for x are 0, 1, –1, 2
Resulting in f(x) values 1, 2, 0, 3
Resulting in pairs (0, 1), (1, 2), (–1, 0), (2, 3)
Graphing the pairs is depicted as:
Is this function continuous at point (1, 2)?
(1 is in the domain set and 2 is in the range set.)
Take the limit as x gets close to 1 and determine whether f(x) gets close to 2.
Limx→1(1 + x)
If x = 1.10, then f(1.10) = 1 + 1.10 = 2.10
If x = 1.01, then f(1.01) = 1 + 1.01 = 2.01
If x = 0.90, then f(0.90) = 1 + 0.90 = 1.90
If x = 0.99, then f(0.99) = 1 + 0.99 = 1.99
Yes, as x gets closer to 1, f(x) gets closer to 2.
Therefore, the function is continuous at this point.
• A method used to visualize whether a function is continuous involves use of symbols such as Epsilon ε and Delta δ to define regions in question on the X and Y axes of the graph of a function. Consider the limit of the function f(x).
Limx→af(x) = L
Where:
The limit exists.
ε represents an error tolerance allowed for L.
δ represents the distance that x is from x = a.
In this method, a value for ε is chosen and δ(ε) is the result. (Or a value for δ is chosen and ε(δ) is the result.)
If Limx→af(x) = f(a) = L and the limit exists, such that:
L + ε = f(a) + ε
L = f(a)
L – ε = f(a) – ε
Then the following is true:
f(a) + ε > f(x) > f(a) – ε or equivalently, |f(x) – L| < ε
For every chosen number ε where ε>0, there is a positive number for δ(ε) that results.
For a chosen f(x), x must be within a – δ and a + δ, such that:
a – δ < x < a + δ or equivalently, 0 < |x – a| < δ
Also, f(x) is a continuous function at x = a if: f is defined on the open interval containing “a”, the Limx→af(x) = f(a), and the limit exists.
• A polynomial function, in general, is continuous everywhere and its graph is a continuous curve. However, a polynomial function is not continuous if there are ratios of polynomial functions and the denominator is zero. The point where a denominator is zero, the function is discontinuous.
• A quadratic function is a type of polynomial function and is continuous everywhere. Taking the limit:
Limx→af(x) = Limx→a(Ax2 + Bx + C) = f(a)
• A function that is not continuous may be discontinuous at a single point. For example, the following function is continuous except at a = 0 where it is discontinuous: (Remember 1/0 is undefined.)
Limx→af(x) = Limx→a1/x = 1/a
• The following is a graph of a function that is discontinuous at a point:
One point can be inserted to make the graph continuous.
• The following is a graph of a function that is discontinuous at more than one point:
One point cannot be inserted to make the graph continuous; it has a “jump.”
• In summary:
If Limx→af(x) exists and is equal to L, but f(x) does not exist when x = a so that f(a) ≠ L, then the graph of f(x) is discontinuous at the point x = a. In this case there is only discontinuity at a single point.
If Limx→af(x) does not exist because as x approaches a from either x>a or x<a, the value of f(x) approached from x>a is different from the value of f(x) approached from x<a, then the graph of f(x) is discontinuous and has a jump at point a = x.
If Limx→af(x) does not exist because as x approaches a, the absolute value of f(x), |f(x)|, gets larger and larger, then the graph is “infinitely discontinuous” at x = a.