Trevor Siu Limits Finding limits is a very useful skill in Calculus. Finding limits is the method of evaluating functions as they approach a number. This is very useful when a function, F(x), has an asymptote at x. However, it is even more useful to find the value of F(x) when x has an asymptote as x->∞, such as this example below. So how do we do this?! There are a few ways to evaluate limits as x->∞, three will be discussed in this worksheet. 1. Graphically The first way to evaluate a limit as x->∞, is to look at the graph of the function, and see what number the graph is approaching as x gets larger and larger. As you can see, the graph is approaching Y=2 as X gets larger in the positive X direction. This means that limx->+∞=2, that when X=∞, then Y=2, however during the period before X=∞, Y does not equal 2. They are not mutually exclusive, however. Just because X doesn’t equal infinity it doesn’t necessarily mean that Y cannot equal 2, just that over the long term, it doesn’t. Having a limit doesn’t necessarily mean that the graph and the number that it is approaching never intersects. As you can see, the graph of F(x) intersects with Y=0 several times, yet limx->∞F(x) is 0. However, it may be difficult to determine the value that Y is, as X approaches ∞, so it may be necessary to use a different method to evaluate the limit, thus the next method. 2. Using a Table The second method to evaluate a limit as X approaches ∞, is to use a table of values generated X 1 10 100 1000 10000 100000 1000000 Y 7 9 9.9 9.99 9.999 9.99999 9.9999999 As you can see, as the X values increase in the positive infinity direction, the value of Y approaches 10, so that means that limx->∞F(x)=10. You can use this method to evaluate limits by putting F(x) into your calculator, in the Y= window, and then pushing the table button. As you plug in values of X that get larger and larger, than you will see the limit start to emerge. The limit will be the number that all the Y values are approaching as X goes to ∞. 3. Algebraically This is the most fun way to evaluate limits as X->∞, which uses all your basic algebra 1 skills (see? They are important…) An example is this: So, start off with the original equation, y=x^3-x^2-3x, and then divide by the highest variable, which in this case is x^3. As you divide by x^3, you get (1-(1/x)-(3/x^2)), so the resulting equation is (x^3)(1-(1/x)-(3/(x^2))). You need to put the (x^3), in order to keep the original equation. The easiest way to explain it, is that you can multiply the original equation, y=x^3-x^2-3x, by 1. One can be any number over any number, so we say (x^3/x^3), and multiply the equation by it. We divide all the terms by the bottom x^3, and we have an x^3 left over, thus (x^3)(1-1/x^-3/x^2). We can evaluate this by plugging -∞ into x. Thus 1/x goes to 0, because the denominator is getting larger in the negative infinity direction, and the fraction approaches zero. 3/x^2 goes the same way, thus the equation becomes (-∞)(1-0-0), thus (-∞)(1)=(-∞). And, there you have it! You are now Calculus champs. Go rule the world, and remember to know your limits!