Unit 5 Limit Analysis

A guide for current PreCalculus students studying for the final.

A. Aggarwal et J. Yang (2020)


An Introduction to Limits

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.



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)

End Behavior of Rational Functions

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

If you are to remember anything about determining the end behavior 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 the end behavior.

There are two forms of end behavior of a rational function: horizontal and non-horizontal.

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

-50/0, which is

undefined

, x cannot be equal to 1 because such an input value would create a fraction

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.

Understanding limits in a graph

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.







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.