As the argument (the independent variable or input) of a function approaches a particular value, the dependent variable approaches some other value called the limit. The important word here is approaches. When finding a limit, we’re interested in what happens to the func-tion as the argument gets closer and closer to a certain value without actually reaching it.
Limit of an infinite sequence
Let’s look at an infinite sequence S that starts with 1 and then keeps getting smaller:
S = 1, 1/2, 1/3, 1/4, 1/5, . . .
As we move along in S from term to term, we get closer and closer to 0, but we never get all the way there. If we choose some small positive number r, no matter how tiny, we can always find a number in S (if we’re willing to go out far enough) smaller than r but larger than 0. Because of this fact, we can say, “If n is a positive integer, then the limit of S, as n gets endlessly larger, is 0.” We write this symbolically as
Limn→∞ S= 0
The expression “n→ ∞” translates to “as n approaches infinity.” When talking about limits, mathematicians sometimes say “approaches infinity” to mean “gets endlessly larger” or “gets
arbitrarily large.” This expression is a little bit obscure, because we can debate whether large numbers are really closer to “infinity” than small ones. But we’ll often hear that expression used, nevertheless.
When talking about this sequence S, we can also say, “The limit of 1/n, as n approaches infinity, is equal to 0,” and write
Limn→∞ 1/n= 0
Limit of a function
Now let’s think about what takes place if we don’t restrict ourselves to positive-integer argu-ments. Let’s consider the function
g (x)= 1/x
and allow x to be any positive real number. As x gets larger, g (x) gets smaller, approaching 0 but never getting there. We can say, “The limit of g (x), as x approaches infinity, is 0,” and write
Limn→∞ g (x)= 0
This is the same as the situation with the infinite sequence of positive integers, except that the function approaches 0 smoothly, rather than in jumps.
Are you a nitpicker?
Let’s state the above expression differently. For every positive real number r, there exists a positive real numbers such that
0 < g (s) < r Also, if t is a real number larger than s, then
0 < g (t) < g (s) < r
Think about this language for awhile. It’s a formal way of saying that as we input larger and larger positive real numbers to the function g, we get smaller and smaller positive reals that “close in” on 0. This statement also tells us that even if we input huge numbers such as 1,000,000, 1,000,000,000, or 1,000,000,000,000 to the function g, we’ll never get 0 when we calculate g (x). We can’t input “infinity” in an attempt to get 0 out of g, either. “Infinity” isn’t a real number!
Are you confused?
If the notion of “closing in on 0” confuses you, look at the graph of the function g for large values of x. As you move out along the x axis in the positive direction, the curve gets closer and closer to the x axis, where Concept of the Limit 21
g (x)= 0. No matter how close the curve gets to the axis, you can always get it to come closer by moving out farther in the positive x direction, as shown in Fig. 2-1. But the curve never reaches the x axis.
Sum rule for two limits
Consider two functions f (x) and g (x) with different limits. We can add the functions and take the limit of their sum, and we’ll get the same thing as we do if we take the limits of the functions separately and then add them. Let’s call this the sum rule for two limits and write it symbolically as
Limx→k [ f (x)+g (x)] = Lim
x→k f (x)+ Lim
x→k g (x)
where k, the value that x approaches, can be a real-number constant, another variable, or
“infinity.” This rule isn’t restricted to functions. It holds for any two expressions with defin-able limits. It also works for the difference between two expressions. We can write
Limx→k [ f (x) − g (x)] = Lim
x→k f (x) − Lim
x→k g (x) g(x)
g(x) = 1/x
x
x
x
Axis Curve
Keep
going
Keep
going
Magnify
Figure 2-1 As x increases endlessly, the value of 1/x approaches 0, but it never actually becomes equal to 0.
In verbal terms, we can say these two things:
• The limit of the sum of two expressions is equal to the sum of the limits of the expressions.
• The limit of the difference between two expressions is equal to the difference between the limits of the expressions (in the same order).
Multiplication-by-constant rule for a limit
Now consider a function with a defined limit. We can multiply that limit by a constant, and we’ll get the same thing as we do if we multiply the function by the constant and then take the limit. Let’s call this the multiplication-by-constant rule for a limit. We write it symbolically as
c Limx→k f (x)=Lim
x→k c [ f (x)]
where c is a real-number constant, and k is, as before, a real-number constant, another vari-able, or “infinity.” As with the sum rule, this holds for any expressions with definable limits, not only for functions. In verbal terms, we can say this:
• A constant times the limit of an expression is equal to the limit of the expression times the constant.
Here’s a challenge!
Determine the limit, as x approaches 0, of a function h (x) that raises x to the fourth power and then takes the reciprocal:
h (x)= 1/x4 Symbolically, this is written as
Limx→0 1/x4
Solution
Asx starts out either positive or negative and approaches 0, the value of 1/x4 increases endlessly. No matter how large a number you choose for h (x), you can always find something larger by inputting some x whose absolute value is small enough. In this situation, the limit does not exist. You can also say that it’s not defined.
Once in awhile, someone will write the “infinity” symbol, perhaps with a plus sign or a minus sign in front of it, to indicate that a limit blows up (increases without bound) positively or negatively. For example, the solution to this “challenge” could be written as
Limx→0 1/x4= ∞ or as
Limx→0 1/x4= + ∞
Concept of the Limit 23