Limits Involving Infinity. Example Problem 3
When approaching a limit, we should evaluate the function numerator and denominator to see what the limit is. In this case, we get 12 IS divided by 0. This is not an indeterminate form; it cannot be factored, expanded, or otherwise manipulated. Because you cannot divide t number by zero, the answer is undefined. When you are approached with an equation that is divided by zero, be aware that you may be dealing with a vertical asymptote. This occurs when both sides of the equation point upward, indicating that the limit is positive infinity. There are three possible cases when a number is divided by zero: positive infinity, negative infinity, or no limit. lim_(x->4) 3x/(x-4) -> 12/0 #/0 x->4^- | x->4^+ x=3.9 | x=4.1 + | + _______________________ – | + -> -infinity |-> infinity [Graph] DNE So, since we already know the answer to this problem, we just need to find out whether it is positive or negative. We’ll do this by doing a little fuzzy math. To explore the limits as we approach 4 from both sides, let’s begin by looking at the left-hand side of the equation. As we approach the number 4 from the left, choose any number between 3 and 4. Someone might be inclined to choose 3 because it is to the left of 4; however, 3 is just a little bit too far away from 4. There really is an infinite number of numbers between 3 and 4. Choose something closer. 3.9 should be more than sufficient. 3 × 3.9 is a positive number. Because the numerators are the same, we know that 12 is correct. Let’s examine the denominator. The denominator, 3.9 minus 4, gives us a negative number. So the denominator is negative. A positive divided by a negative is a negative. So from the left-hand side, we are approaching infinity. As shown here. the graph will look like this. Let’s now look at the right-hand side of 4. We’ll use a 1/10 difference once again, so 4.1 seems sufficient. Numerator still going to be positive. Nothing’s going to change there. The denominator is 4.1 minus 4, which is a positive number. A positive divided by a positive is a positive number, sO the right-hand side will approach positive infinity. So the left side approaches negative infinity, and the right side approaches positive infinity. When the two one-sided limits de not agree with each other, we say the limit does not exist.
a
