Example Problem 3. Limits Involving Infinity

Limits Involving Infinity. Example Problem 3 When approaching a limit, we should evaluate the function numerator and denominator tosee what the limit is. In this case, we get 12 divided by 0. This is not an indeterminate form; it cannot be factored,expanded, or otherwise manipulated. Because you cannot divide a number by zero, theanswer is undefined.When you are approached with an equation that is divided by zero, be aware that you maybe dealing with a vertical asymptote. This occurs when both sides of the equation pointupward, indicating that the limit is positive infinity.There are three possible cases when a number is divided by zero: positive infinity, negativeinfinity, or no limit. So, since we already know the answer to this problem, we just need to find out whether it ispositive 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-handside of the equation. As we approach the number 4 from the left, choose any number between 3 and 4. Someonemight be inclined to choose 3 because it is to the left of 4; however, 3 is just a little bit too faraway from 4. There really is an infinite number of numbers between 3 and 4. Choosesomething closer.3.9 should be more than sufficient. 3 × 3.9 is a positive number. Because the numerators arethe 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 theleft-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 4.1seems sufficient. Numerator still going to be positive. Nothing's going to change there. Thedenominator is 4.1 minus 4, which is a positive number. A positive divided by a positive is apositive 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 do not agree with each other, we say the limit does not exist.