Lecture2: Limits
Notation
The symbol means "approaches" in calculus when we discuss limits.
x2 is read "x approaches 2".
As superscripts, the symbol + means 'from the right' (or positive direction), and ¯ means "from the left" when we discuss limits. Thus
x2+means "x approaches 2 from the right".
x2¯ means "x approaches 2 from the left".
Graphical and Numerical Approaches
Consider the function . Note that the function is undefined at the value
x = 1, owing to the fact that the denominator x - 1 is equal to zero there. Otherwise, f (x) behaves like the linear function x + 1, since
So the graph of f(x) would be the same as the graph of the line y = x + 1, except at x = 1, where there is a 'hole' in the function:
figure 1
What we would like to do is examine the behavior of the function near x = 1, and draw some conclusions. There are too ways we might go about this. One is to simply check out values of f(x) for x's near x = 1, perhaps in a table:
x | .75 .9 .99 .999 1 1.001 1.01 1.1 1.25
f(x) | 1.75 1.9 1.99 1.999 2.001 2.01 2.1 2.25
Take a look at the top row from left to right until we get to 1 - the values go up slowly from the left toward 1: .75, .9, .99, .999, and underneath them the (y-) values of f(x) increase toward what appears to be 2: 1.75, 1.9, 1.99, 1.999. Thus, we feel comfortable saying that as
x 1¯, f(x) 2.
Now look in from the right side at the top row - the values of x decrease: 1.25, 1.1,1.01, 1.001. The corresponding y-values of f(x) also decrease: 2.25, 2.1, 2.01, 2.001, seemingly "zooming in" on the value 2. We say that as
x 1+, f(x) 2.
And so it seems that as x approaches 2 from either side, our function has
y-values that "zoom in", or approach the value y = 2. We characterize this behavior as a limit of f(x) and introduce the following notation:
This is read "the limit of f as x approaches x is 2", and generally:
is read "the limit of f as x approaches c is L". The formal definition of limit will be given in a later section.
Our deduction of the limit in this case was an example of the use of a numerical approach, but this is not the only technique available for finding said limit. For instance, this limit could have been determined by inspecting the graph of the function:
1
Note that as we approach the x-value of 1 from either side, the function "zooms in on" the hole at (1,2), and so the y-values (f(x)) are approaching y = 2.
Ex:
(1) Use the graphical method to find
->There may be cases where the limit of a function does not exist at a given value of x. There are distinctive geometric features of such functions. Let's look at some key examples:
Here, as we approach 1 from the left, the function is 'zooming in' on y = 0, but as we approach from the right, the function is zooming in on y = 1. In such cases, where the function approaches a different value from left and right, we say the limit 'does not exist', or
Note the behavior of the graph of the function where the limit DNE - it has a 'jump' .
Ex: (2) Graph the piecewise function and give if it exists.
Another case where a limit may not exist is at a vertical asymptote, such as a rational (or trig) function at a value where the denominator is undefined.
For example, the function has a vertical asymptote at x = 2, and the reason that the limit DNE there is because the function does not 'zoom in' on a finite value from either side. We call these 'infinite limits'.
Thus far, we have looked at the concept of limits in a natural and non-technical way, thinking of them in terms of 'approaching' and 'zooming in'. There is a technical definition of the limit, described on p. 52. I will write it differently than the text, using the logical symbols = 'for every', = 'there exists', = 'so that', ='implies'.
means
Ex: Ex. (7), p. 53.
1.3
->That's not what I would describe as fun, that Ex 7, p. 53. It's a tough way to find a limit, and it would be nice if we could find some more convenient way to assess them. And in fact, there is, and the methodology we will use to solve the problem is similar to an approach we will use to crack a whole range of problems we will confront in Calculus.
->What we hope to do is find a set of rules for finding limits of functions. The key is: to consider what types of functions we would like to find limits for. The functions you have seen to now include: polynomials, rational functions, some piecewise functions and trigonometric functions. Next, we need to formulate the aforementioned rules for assessing limits of these functions, and said rules come in the form of theorems. Section 1.3 lists these theorems. I will discuss some of them and show how we put them to use.
Let's take a look at these 3 rules, which are listed on p. 59.
rule: example:
The last example is telling - by combining the rules, we can evaluate the limit of the polynomial function by simply plugging in x = c, and this is, in general, true, as we see in theorem 1.3 on p. 60 if p(x) is a polynomial, then
, that is, we can simply plug c into the function for x to find the limit.
Ex: (1) Find .
Because a rational function is just the quotient of two polynomials, and by the limit rule:
, we can assess the limits of rational functions by just plugging in
x = c as well: , as long as Q(c) is not equal to 0.
Ex: (2) Find
Theorem 1.4 lays out a similar condition for root functions, 1.5 for 'composed' functions (functions with other functions 'embedded ' within them). And theorem 1.6 says that we can also evaluate trig functions by simply plugging in x = c, as long as the trig function exists at x = c.
Ex: Find the limit:
(a)
(b)
(c)
This is good news, and greatly simplifies the evaluation of limits for many of the functions we are interested in studying.
-> But there still may be some problems -
Ex: Ex (7) , p. 63,
Ex (8), p. 64
Finally, there are 2 special trigonometric limits presented on p. 65, which may be needed in disguised form :