Remark: Do not treat as ordinary numbers. These symbols do not obey the usual rules of arithmetic, for instance, + 1 =, - 1 =, 2, etc.

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1 Limits and Infinity One of the mysteries of Mathematics seems to be the concept of "infinity", usually denoted by the symbol. So what is? It is simply a symbol that represents large numbers. Indeed, numbers are of three kinds: large, normal size, and small. The normal size numbers are the ones that we have a clear feeling for. For example, what does a trillion mean? That is a very large number. Also numbers involved in macro-physics are very large numbers. Small numbers are usually used in micro-physics. Numbers like are very small. Being positive or negative has special meaning depending on the problem at hand. The common mistake is to say that - is smaller than 0. While this may be true according to the natural order on the real line in term of sizes, - is big, very big! So when do we have to deal with and -? Easy: whenever you take the reciprocal of small numbers, you generate large numbers and vice-versa. Mathematically we can write this as: (where you must determine the sign +/-) Note that the reciprocal of a small number is a large number. So size-wise there is no problem. But we have to be careful about the positive or negative sign. We have to make sure we know whether a small number is positive or negative. 0+ represents small positive numbers while 0- represents small negative numbers. (Similarly, we will use e.g. 3+ to denote numbers slightly bigger than 3, and 3- to denote numbers slightly smaller than 3.) In other words, being more precise we have Remark: Do not treat as ordinary numbers. These symbols do not obey the usual rules of arithmetic, for instance, + 1 =, - 1 =, 2, etc. Valid Infinity Arithmetic 4 5* = -2* = - Indeterminate Forms,, - More to come later (math151).

2 -4 But, beware: - is Indeterminate. I. Determining the behavior of a function on either side of a Vertical Asymptote. Recall: The zeros of the denominator of a function are either going to be the location of a vertical asymptote or the location of a hole in the graph. Example: At what values of x are the vertical asymptotes located? At what values of x where there be a hole in the graph? f(x) =. We will begin with a very simple rational function and examine the behavior around the vertical asymptote. When x 3, the x So, we must check on either side of 3. Let x. This can be read: as x approaches 3 from the left, the denominator approaches 0 negatively or with negative values, so approaches negative infinity. Let x Let x. This can be read: as x approaches 3 from the right, the denominator approaches 0 positively or with positive values, so approaches positive infinity.

3 Note that when x gets closer to 3, then the points on the graph get closer to the (dashed) vertical line x = 3. Such a line is called a vertical asymptote. For a given function f(x), there are four cases, in which vertical asymptotes can present themselves: (i) (ii) (iii) (iv) ; ; ; ; ; ; ; ; While the denominator is the same as the previous example, there is a function in the numerator that will affect the behavior as x approaches 3. First graph on Desmos.com. Find and For this function we can use properties of limits along with the basic graph of y =

4 = = Find and Find and Find and Next we investigate the behavior of functions when x. We have seen that. So for example, we have In the next example, we show how this result is very useful. Example: Consider the function f(x) = We have

5 = = = 2 Note that when x gets closer to (x gets large), then the points on the graph get closer to the horizontal line y=2. Such a line is called a horizontal asymptote. In particular, we have for any number a, and any positive number r, provided x r is defined. We also have For -, we have to be careful about the definition of the power of negative numbers. In particular, we have

6 for any natural number n. If we wish to determine the behavior of the function as x, we can divide both the numerator and denominator by the greatest power of x that occurs in the denominator, which is x 4. We have:. So we have Find the vertical and horizontal asymptotes for the graph of f(x) = Example: Consider the function f(x) =

7 We have = And working on the denominator: 3x + 1 = x(3 + and then = When x goes to, then x > 0, which implies that x = x. Hence When x goes to -, then x < 0, which implies that x = -x. Hence Remark. Be careful! A common mistake is to assume that. This is true if x and false if x < 0.

8 Rational Functions and Horizontal Asymptotes R(x) = Case 1: If the degree of the numerator is less than the degree of the denominator, the HA is y = 0. Case 2: If the degree of the numerator is the same as the degree of the denominator, the HA is y =. Case 3: No Horizontal Asymptote. The rational function is approaching infinity either positively or negatively. You must find the limits as x and as x to determine the end behavior. Limits: Why are limits useful?? For mathematicians they are fundamental in the development of the derivative and the integral, the two primary mathematical structures in calculus. It is with the use of limits that we can take mathematics from discrete values to continuous values. The derivative as the slope of the tangent line to a graph & the definite integral as the area under the graph of a function are two classic applications of the primary structures in calculus. The derivative as the slope of the tangent line is the first we will see. For us, we will cover this Monday. Let us consider a concept that used limits in your previous math class. Example 1: An example that may be somewhat familiar is finding the sum of a geometric series such as = We derived the formula for a geometric series (provided r < 1) in math 96 and found so if we want to find the sum of the entire infinite series, a + ar + ar = = provided r < 1 Applying this to the repeating decimal , we can see that a = 0.24, r = 0.01 which is less than 1 in absolute value, so that = = Example 2: Suppose that a very large tank initially contains 200 gallons of a saline solution with 10 lb of salt. At 10 a.m. a solution is pumped into the tank at 4 gallons/hour that contains 5 lb salt/gallon. Find the following. a) The amount of solution at any time t,

9 b) The amount of salt in the tank at any time t, c) The salt concentration at any time t, d) What is the limit of the salt concentration going to be as t?