Document Sample

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!

DOCUMENT INFO

OTHER DOCS BY tsiu

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.