In most mathematics, the concept of a limit is extremely important. Taking the limit of a sequence will tell us what happens at an infinite point of the sequence; it would be impossible to do this without limits, since we cannot do anything infinite times. Taking the limit of a function at a certain point is also significant - it gives us a lot of precise information about a function around a given point. In short, the limit is tremendously useful and is used in almost every branch of mathematics. For clarity's sake, we will begin with the limit of a sequence.
Suppose we have a sequence of numbers 2, 5, 8, 11, 14, 17, ... We can see that there is a pattern that describes the sequence of numbers. Knowing that pattern, we can give an explicit expression for the sequence: If we index 2 as the zeroeth term of the sequence, then the nth term is 2 + 3n. Mathematically, we write
an represents the nth term of the sequence. The limit of the sequence is the number the sequence approaches if we look at an infinite number of terms. We can see that the more terms we take, the greater the number gets, so if we take an infinite number of terms, the sequence goes to infinity. It is written mathematically like this:
This is an easy example, and it can be figured out logically. However, we need a methodical way of solving limits. In general, we first try plugging in ∞ for n, and evaluating the limit according to a few rules.
Rules For Computing Limits
To begin with, we must start with the fundamental rules of limits.
1. The Constant Rule
When we take the limit of a constant, non-changing function, the limit will simply be that constant. For example, suppose an = 4 no matter what n we choose(note that the function is just a single number; this is what we mean by a constant function). This sequence would just be 4, 4, 4, 4, 4, ... Then, if we took the limit of this function, , it would always be 4.
2. The Multiplication Rule
If two sequences have limits that exist, then the limit of the product is the product of the limits. Suppose we know that and, for a different sequence bn, , then we immediately know, by the multiplication rule, that . In this case, the limit of our product, anbn, is equal to the product of the limit of an, A, and the limit of bn, B, otherwise known as the product of the limits. This rule will have more application when we get to limits of functions.
3. The Sum Rule
If two sequences have limits that exist, then the limit of the sum of sequences is the sum of the limits of the sequences. Suppose, again, that we know and . Then, by this rule, we therefore know that . This rule will also have more application when we get to limits of functions, but it is still useful with sequences.
Remember that in order for rules 2 and 3 to be relevant, our limits must exist. If you end up with a limit of ∞ or a non-existent limit, you can not add or multiply it to anything.
Beyond these rules, there are a few very important truths to know when calculating limits.
1/∞ = 0. This is based on the premise that a number with a very large denominator, like 1/999999, is very close to zero.
Also, if our sequence is described by a fraction with powers of n on the top and the bottom,
if the degree of n is higher on top, then the limit is infinity:
if the degree of n is higher on the bottom, then the limit is zero:
if the degree of n is the same in the numerator and denominator, then the limit is the ratio of the leading coeffecients:
In general, since infinity has the quality that a higher power of infinity is uncountably greater than a lower power, we only really worry about the highest powers of n. To see in greater detail the motivation behind these rules and examples, look at the following flash animation:
Limits of Functions
The notion of a limit becomes very useful when we look at
functions. With functions, we take the limit of a function as our
dependent variable, x, approaches a certain x-value. The limit will tell us the value of a
function at a certain point. For example, if we have f(x)
= x over (x+4), then we can take the value of this function at x =
1. limf(x) = 1/5.
We may also learn alot about the function using limits. Through our knowledge of domains, we know that the function is not defined at x = -4. We can check what happens to the function at a discontinuity by taking a limit at this point. limf(x) = ∞. This tells us that the function has an asymptotic discontinuity at x = -4. Graphically, it also tells us that the function approaches either positive or negative infinity.
It may also help to see what happens to the function as the x approaches infinity - this will tell us the behavior for large x-values. Again using the function defined above, if we take the limit as x approaches infinity we find that limf(x) = 1, by using the same rules as limits of sequences. This tells us that along the positive x-axis, the graph of the function will eventually be very close to the line y = 1.
Left-hand and Right-hand LimitsIf we look at the graph of a function like _____________, you may notice that at the discontinuity x = -4, on one side of it the function approaches positive infinity, and the other side of the function approaches negative infinity. At certain points, a function can have different neighborhood on its left and right sides. Because of this, we can take one-sided limits. In our example, we may try taking the left-hand limit at our discontinuity, connotated by. The -4- represents a number infinitely close to -4 but the tiniest bit smaller than it (or graphically, to the left of it). You can think of it as -4.00001 (or -4.0000000000001, etc.). The point is that it is very close to -4 but is clearly on the left-hand side of the limit of -4. We take the limit just as we would normally, but now, the (x+4) term in the denominator is definitively defined as (-4- + 4). Remembering that this is similar to (-4.0000001 +4), we can see that the denominator is now still very close to zero but it is decidedly negative. This tells us that the function approaches negative infinity on the left side. A right-hand limit functions similarly, except we take limf(x).