Limits
Overview
Substitution
Factoring
Conjugate Method
Crazy Graphs
One – Sided
Other Techniques
What is a Limit?
The limit of a function is the value the function approaches at a given value of x, regardless of whether the function actually reaches that value. For an easy example, consider the function
When x=5, f(x)=6. Therefore, six is the limit of the function at x=5, because six is the value that the function approaches as the value of x gets closer and closer to five.
I know it’s strange to talk about the value that a function “approaches.” Think about it this way: If you set x=4.9999 in the function above, then f(x)=5.9999. Similarly, if you set x=5.0001, then f(x)=6.0001.
You can begin to see that as you get closer to x=5, whether you’re approaching it from the 4.9999 side or the 5.0001 side, the value of f(x) gets closer and closer to six.
In this simple example, the limit of the function is clearly six because that is the actual value of the function at that point; the point is defined. However, finding limits gets a little trickier when we start dealing with points of the graph that are undefined.
In the next section, we’ll talk about when limits do and do not exist, and some more creative methods for finding the limit.
When Does a Limit Exist?
General vs. One-Sided Limit
When you hear your professor talking about limits, he or she is usually talking about the general limit. Unless a right- or left-hand limit is specifically specified, you’re dealing with a general limit.
The general limit exists at the point x=c if
1. The left-hand limit exists at x=c,
2. The right-hand limit exists at x=c, and
3. The left- and right-hand limits are equal.
These are the three conditions that must be met in order for the general limit to exist. The general limit will look something like this:
You would read this general limit formula as “The limit of f of x as x approaches two equals four.”
Left- and right-hand limits may exist even when the general limit does not. If the graph approaches two separate values at the point x=c as you approach c from the left- and right-hand side of the graph, then separate left- and right-hand limits may exist. Left-hand limits are written as
The negative sign after the two indicates that we’re talking about the limit as we approach two from the negative, or left-hand side of the graph.
Right-hand limits are written as
The positive sign after the two indicates that we’re talking about the limit as we approach two from the positive, or right-hand side of the graph.
In the graph below, the general limit exists at x=-1 because the left- and right- hand limits both approach one. On the other hand, the general limit does not exist at x=1 because the left-hand and right-hand limits are not equal, due to a break in the graph. In the first graph on the left, left- and right-hand limits are equal at x=-1, but not at x=1.
Where Limits Don’t Exist
We already know that a general limit does not exist where the left- and right-hand limits are not equal. Limits also do not exist whenever we encounter a vertical asymptote.
There is no limit at a vertical asymptote because the graph of a function must approach one fixed numerical value at the point x=c for the limit to exist at c. The graph at a vertical asymptote is increasing and/or decreasing without bound, which means that it is approaching infinity instead of a fixed numerical value.
In the second graph to the left, separate right- and left-hand limits exist at x=1 but the general limit does not exist at that point. The left-hand limit is four, because that is the value that the graph approaches as you trace the graph from left to right. On the other hand, the right-hand limit is negative one, since that is the value that the graph approaches as you trace the graph from right to left. The general limit does not exist at x=1 or at x=2.
Where there is a vertical asymptote at x=2, the left-hand limit is negative infinity, and the right-hand limit is positive infinity. However, the general limit does not exist at the vertical asymptote because the left- and right-hand limits are unequal.
Solving Limits Mathematically
Just Plug It In
Sometimes you can find the limit just by plugging in the number that your function is approaching. You could have done this with our original limit example, the limit as x approaches five of the function x+1. If you just plug five into this function, you get six, which is the limit of the function. To the left is another example, where you can simply plug in negative two to the function to solve for the limit.
Factor It
When you can’t just plug in the value you’re evaluating, your next approach should be factoring. See the factoring example to the left for more information.
Conjugate Method
This method can only be used when either the numerator or denominator contains exactly two terms. Needless to say, its usefulness is limited. Here’s an example of a great, and common candidate for the Conjugate Method.
In this example, the substitution method would result in a zero in the denominator. We also can’t factor and cancel anything out of the fraction. Luckily, we have the Conjugate Method. Notice that the numerator has exactly two terms, the square root of (4+h) and -2.
Conjugate Method to the rescue! In order to use it, we have to multiply by the conjugate of whichever part of the fraction contains the two terms. In this case, that’s the numerator. The conjugate of two terms is those same two terms with the opposite sign in between them.
Notice that we multiply both the numerator and denominator by the conjugate, because that’s like multiplying by one, which is useful to us but still doesn’t change the value of the original function. See the example on the left for more detail about the Conjugate Method.
Remember, if none of these methods work, you can always go back to the method we were using originally, which is to plug in a number very close to the value you’re evaluating and solve for the limit that way.
