Limits+involving+infiniti

 **"ARE YOU READY FOR THIS?!?" - Sir Isac Newton**

= = toc

=Section 2-5: Limits Involving Infinity=

Get your thinking caps on, ladies. This is probably going to hurt.

So lets say there are a bunch of bunnies in the room. Food is occasionally dropped into the room. The rabbits will keep breeding continuously, so the population will grow. The food, however, is limited, so the room can only support so many rabbits until they start to die of starvation. Eventually the population will stabilize when there are just enough rabbits for the amount of food. On a graph this might look a little bit like

The function seems to have a horizontal asymptote around 67 or so. We can express this with limits.

Another example would be if the rabbit population was fed as much as it wanted, and the room expanded accordingly. There would be no limit to the rabbit population and so the function might look a bit like Here the graph is like WOAH positive, and seems to have a vertical asymptote. How on earth do we express this with limits?


 * SO GLAD YOU ASKED**

The First Bit: Limits that equal infinity
First things first:



These might look very complicated but it's actually quite simple. The first part of each means the limit of a function f(x) as x approaches c. That sideways eight after the equals sign is a handy dandy symbol for infinity, positive on the top and negative on the bottom (as denoted by the "-" sign). Basically, we use these two notations to say that a function approaches infinity or negative infinity at some value of x.

GOT IT? awesome

So how can we tell something is approaching infinity? Lets to an example, eh?

**//
 * // Example 1 //** Evaluate each of the following limits.

Solution//**

So we’re going to be taking a look at a couple of one-sided limits as well as the normal limit here. In all three cases notice that we can’t just plug in x=0. If we did we would get division by zero. And if we divide by zero we get  Now, there are several ways we could proceed here to get values for these limits. One way is to plug in some points and see what value the function is approaching. Nice and easy like.

So, here is a table of values of //x//’s from both the left and the right. Using these values we’ll be able to estimate the value of the two one-sided limits and once we have that done we can use the fact that the limit will exist only if the two one-sided limits exist and have the same value.


 * //x// || 1/x || //x// || 1/x ||
 * -0.1 || -10 || 0.1 || 10 ||
 * -0.01 || -100 || 0.01 || 100 ||
 * -0.001 || -1000 || 0.001 || 1000 ||
 * -0.0001 || -10000 || 0.0001 || 1000 ||

From this table we can see that as we make //x// smaller and smaller the function 1/x gets larger and larger and will retain the same sign that //x// originally had. It should make sense that this trend will continue for any smaller value of //x// that we chose to use. The function is a constant (one in this case) divided by an increasingly small number. The resulting fraction should be an increasingly large number and as noted above the fraction will retain the same sign as //x//.

We can make the function as large and positive as we want for all //x//’s sufficiently close to zero while staying positive (//i.e.// on the right). Likewise, we can make the function as large and negative as we want for all //x//’s sufficiently close to zero while staying negative (//i.e.// on the left). So, from our definition above it looks like we should have the following values for the two one sided limits.  Another way to see the values of the two one sided limits here is to graph the function. Again, in the previous section we mentioned that we won’t do this too often as most functions are not something we can just quickly sketch out as well as the problems with accuracy in reading values off the graph. In this case however, it’s not too hard to sketch a graph of the function and, in this case as we’ll see accuracy is not really going to be an issue. So, here is a quick sketch of the graph.  So, we can see from this graph that the function does behave much as we predicted that it would from our table values. The closer //x// gets to zero from the right the larger (in the positive sense) the function gets, while the closer //x// gets to zero from the left the larger (in the negative sense) the function gets.

Finally, the normal limit, in this case, will not exist since the two one-sided have different values.

So, in summary here are the values of the three limits for this example. 

Wow. You can now take a limit that approaches infinity. Way to go! Ready for more? You bet you are.

The Second Bit: Limits as x approaches infinity
Look at this one!

This is the graph of 1/(x+3). Previous to this moment, you might have said "hey, that graph looks like has a horizontal asymptote of 3.

Not any more. For he to-day who learns with me shall be my brother; be he ne'er so vile, This day shall gentle his condition; And gentlemen in England now-a-bed shall think themselves accurs'd they were not here, And hold their manhoods cheap whiles any speaks That learned to-day of limits.

Lets see what the tables tell us, shall we?

We can see that as x increases, 1/x + 3 gets closer and closer to 3 from both high and low numbers. We can write this as
 * x || 1/x + 3 || x || 1/x + 3 ||
 * 10 || 3.1 || -10 || 2.9 ||
 * 100 || 3.01 || -100 || 2.99 ||
 * 1000 || 3.001 || -1000 || 2.999 ||
 * 10000 || 3.0001 || -10000 || 2.9999 ||



and So as x gets infinitely positive or infinitely negative, the limit of 1/x + 3 approaches 3. Neat, huh?


 * Oscillating Function. **

 huffingtonpost.com The graph above of the oscillating function is approaching a limit as x approaches infinity. In the graph of an oscillating function pictured above, the function does not appear to be approaching a limit. A graph of an oscillating function is approaching a limit if y(x) can be kept arbitrarily close to y(c), whilst c approaches negative or positive infinity. Exmple: y(x) = 5 + 4(.7)x cos ( x) As you can see from the graph lim **x**→∞ y(x) = 5.

This is because L = lim **x**→∞ y(x) if: -for any > 0 (no matter how small) -there is a number D > 0 such that -if x> D, -then y(x) is within E units of L. The larger x gets (approaches infinity) the closer y(x) gets to 5, even though it may oscillate around 5, it still stays within E units of L.