Unit 5 Limit Analysis

Given a function f(x), a limit expresses the value of f(x) as x approaches a certain value. A limit is expressed with the following format for a function f(x): (see right)

α can be any value from -∞ to ∞ included
f(x) is the function
β is value of the limit

Verbally, a limit can be expressed as follows: f(x) as x approaches a.

What we've explained above are called two-sided limits, meaning f(x) approaches β as x approaches α from both the positive and negative side. However, there exist one-sided limits as well.

A one-sided limit is the value the function approaches as the x-values approach the limit from one side only. These will usually show up in graphs of piece-wise or non-continuous functions.

Left-Handed Limits

x approaches α from the left (x increases to get to α)

Right-Handed Limits

x approaches α from the right (x decreases to get to α)

End Behavior

A limit that describes a graph's end behavior describes the graph of a function as the inputs of f(x) reach positive and negative infinity.

For example, for the limit (see right), as x approaches ∞, or x→∞, or x becomes unbounded positively, the denominator (x-1) increases infinitely (becomes unbounded positively) as well. -50 divided by an increasingly large number results in a fraction getting infinitely smaller, or approaching 0.

The (–) sign above the 5 represents that the fraction is approaching 0 from the negative side.

In conclusion, as the fraction approaches 0^– when x→∞ it becomes insignificant. 0^– + 5 equals 5^–. (see left)

Limits with Rational Functions

Here, we are trying to assess the limits behavior of a rational function f(x)/g(x).

If you are to remember anything about determining the limits of a rational function, remember this: the term with the greatest exponent in both the numerator—f(x)—and the denominator—g(x)—outweigh every other term, so they play the biggest role in determining the limit.

An easy way to conceptualize and determine the end behavior of a rational function is to factor out the dominant term in both the numerator and denominator (because that's the only term that really matters).

As x approaches infinity or negative infinity, the the dominant term rules over the other terms because it grows faster than the other terms. We will outline some examples of the limits of different functions as x approaches infinity.

There are three forms of limits of a rational function: horizontal, vertical, and non-vertical (slant).

Horizontal Asymptotes

These occur when the exponent of the largest term (the degree) of the numerator is less than or equal to the degree of the denominator. There are two subcases:

Horizontal Asymptote at y = 0

If the degree of the numerator is less than the degree of the denominator, the limit of the function (and the horizontal asymptote) is 0.

Horizontal Asymptote at y ≠ 0

If the degrees of the numerator and denominator are the same, the limit of the function (and the horizontal asymptote) is the ratio of the leading coefficients.

Vertical Asymptotes and Discontinuities

A discontinuity is formed when an input value cannot be included in the domain since the output would be undefined.

In the function (see right) , x cannot be equal to 1 because such an input value would create a fraction -50/0, which is undefined because one cannot divide by 0.

A limit can be used to evaluate the f(x) values as x approaches 1 (see right). This limit would express the shape of the graph as x approaches 1 from the right, or from the values greater than one (as denoted by the 1^+.)

As x approaches 1, or x→1, the value of the fraction becomes infinitely large, or unbounded positively, as 50 is being divided by an increasingly very small positive number; however, since the fraction is negative, it approaches negative infinity, or becomes unbounded negatively. The +5 is insignificant in this scale; thus, (see left).

For the that limit would express the shape of the graph as x approaches 1 from the left, or from the values less than one (as denoted by the 1^-), (see right).

The only difference is that the value of 50/(x-1) becomes infinitely negative, or unbounded negatively, however, as the fraction is negative, the value of the fraction becomes infinitely positive, or unbounded positively. Again, the +5 is insignificant; thus, (see left).

The unbounded qualities of these limits that evaluate the shape of a function as x approaches a certain number, rather than infinity (like with end behaviors), signify a vertical asymptote.

Other Asymptotes

Slant/Oblique Asymptote

If the degree of the numerator is exactly 1 more than the degree of the denominator, the limit of the function (and the asymptote) is the linear equation obtained by dividing the numerator by the denominator.

Polynomial Asymptote

If the degree of the numerator is more than 1 more than the degree of the denominator, the limit of the function (and the asymptote) is the polynomial obtained by dividing the numerator by the denominator.

Understanding Limits in a raph

Methods of Evaluating (Finite) Limits

Tables

Consider the following limit (see right):

Clearly, we cannot simply substitute in 2 for x (as the function is indeterminate for x = 2).

Pros

Intuitive
Relatively easy (just plug in some numbers)
Can be used even when graphical utilities aren't available or you cannot simplify the function and solve it algebraically

So instead of solving for x = 2, let's create a table of values for x and f(x) as x approaches 2. (see left)

Note how this is basically guessing-and-checking for values (on either side) of x as it gets closer and closer to 2. Here we see that the limit of f(x) as x approaches 2 is going to be 3.

Cons

The answer you get is an approximation
Method can be inefficient
You may have to be careful of one-sided limits (the table will only approach the value from one side)

Graphs

This method is similar to the table method: a graph helps us approximate a limit by allowing us to estimate the finite y-value we're approaching as we get closer and closer to some x-value. But instead of setting up a table, we draw a graph representing the function and estimate the limit based on that.

Consider the same limit as before, except now represented on a graph (see right):

Pros

Easy to see visually what the limit is
Can quickly inspect the limit
Useful in needing to identify a certain type of discontinuity

Cons

The answer you get is an approximation (even a graphing calculator's resolution suffers at some scale)
Drawing a graph may be tedious and difficult (maybe you don't have a calculator)

Algebra

To use algebra, we simply have to take a function that we cannot take the limit of (as is) and convert it into one where we can.

Identifying Holes:

A hole exists if a same factor is in both the numerator and denominator. The x-value of the hole is that at which the factor equals 0. The limit as the function approaches this specific x-value can be found by plugging in the x-value into the reduced function.

You may just have to factor the function and simplify from there. Then you can simply substitute in the value of x for which you are trying to find the limit.

But sometimes, you may have to finesse the problem a little more cleverly. For example, above, you have to multiply the numerator and denominator by the conjugate of the numerator. Still, the foundation of solving this problem lies in finding the limit of a reduced function .

Pros

Quicker than the other two methods (provided you know how to analyze the problem and you have practiced)
The answer is the exact limit, not an approximation

Cons

Perhaps not as intuitive
You have to be confident in your abilities and sometimes creative in your problem-solving (which may require a little more preparation beforehand)
Doesn't work for every function

Note that using algebra is optimal for rational functions, so if you come across a problem like the one on the right, you might want to use another method like creating a table.

Spoiler alert: there is a finite answer.