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Limits and all that silly stuff

Problem Sets (No Popups - Problems | Answers)

Chat about Calculus

Limits are the heart of calculus. You will learn them now, understand in the next section how calculus uses them behind the scenes, and never see them again. The important lesson to take with you is how limits are graphed. For the rest of this course, and beyond, graphs will be intuitive if you understand how the limits work.

Two things you will need to know: 

1 - ∞ represents "infinity". Since infinity is not an actual number, you cannot do real calculations with it. In limits, we often question what happens when something approaches infinity, but not actually reaching it. Infinity is the limit of the equation. This is the method by which we sneak infinity into the calculations. 

 
 

2 - The format of a limiting equation is "lim x --> a   y = equation". This is read as: As x approaches the constant 'a', what does the equation go to?" In the problem, there are two limits involved, and throughout the section you will have look at both of them. There is a limit of x, and a limit of y (which is limit of equation). The problem explicitly states where x goes to. As x approaches that number, the whole equation will approach another. That is the other limit; that is where y goes to. y is equal to the whole equation. I will now start with basic problems. Grasp each before you move on.

3 - Limits of x are on the horizontal axis, and limits of y are on the vertical axis. This should be obvious, but everyone mixes the two up some time or another. Especially with asymptotes. You will learn about asymptotes soon.

Simple Limits

A limit says that x is approaching a number, ranging from negative infinity to positive infinity. 

Lim x-->a   y = 3x + 4         y = 3a + 4

In that case, x was going to a, so you plug a into the equation. Simple enough. It comes out the same as x = a.

Lim x-->0   y = 3x + 4         y = 3(0) + 4 = 4

That equals 4, because the other term is approaching 0.

Lim x-->∞    y = 3x + 4        y = 3(∞) + 4 = ∞ + 4 = ∞

This is where it begins to get more complicated. As the first term approaches infinity, so does the entire function. Even bigger than infinity, technically, (infinity plus four) but that is infinity. Infinity is not a specific number, so anything that just keeps on getting larger without end is said to be going to infinity. Note that when we say the equation is going to a number, or infinity, it is referring to the y-axis of the equation, as x approaches the said number. In this example, both x and y go to infinity.

In the following picture, however badly drawn, as x approaches 0 from the positive side, the y-axis approaches negative infinity. This is an example of how when we say that this function is going to negative infinity we are referring to the y-axis. If you look at the y value at x = 1/2, you will notice it is far less than x = 1, and x = 1/10 has a y value far less then f(1/2). Incidentally, this picture is for function y = ln x, and it will be discussed below. For more on natural logarithms, you can go to my precal page.

Definition of Asymptote (Courtesy of dictionary.com) - "A line which approaches nearer to some curve than assignable distance, but, though infinitely extended, would never meet it." Basically, a curve gets infinitely close to some vertical line, such as x = 0, in y = ln x. Here we have a vertical asymptote, because the curve approaches a vertical line. In horizontal asymptotes, the curve will approach horizontal lines. The graph of y = 1/x, given at the bottom of this page, has two vertical asymptotes, and two horizontal asymptotes. There will be occasional asymptotes throughout the problems on this page; have fun spotting them. 

Lim x-->3   y = (x2 + x – 12)/(x – 3)

This is not simple. If you plug in 3 then the equation is undefined, because it goes to 0/0. What you can do however is factor out the x-3. In problems like this you can usually factor out. It becomes:

Lim x-->3   y = (x – 3)(x + 4)/(x – 3)        y = x + 4        y = 7

I canceled out the (x - 3) and it turns out that as x approaches 3, the y-axis of the equation goes to 7. Notice however that it NEVER equals seven! It gets infinitely close. There is a gap in the equation at the coordinates (3,7).

 

Dealing with x going to infinity

When x goes to infinity, you must look to see if x is in the numerator (or if it's not a fraction, same thing), if x is in the denominator, or both. I will cover these three scenarios.

If x is either in the numerator (top) of a fraction or in a regular number, the curve is going to go higher on the y-axis as x approaches infinity. An example is the simple equation y = x2 As x gets bigger, so does y.

Another scenario is where x is on the bottom of a fraction. A simple case is y = 1/x.  Here, as x gets very big, the equation becomes very small. (or say that y gets small) When x is a million, the equation equals one millionth, an extremely small number. So in this case, the limit as x goes to infinity is 0, because it is approaching that. It will never actually be zero, but that doesn’t matter in a limit against infinity. Regardless, infinity is impossible, since it does not imply reality, but in the world of infinity, this equation actually does equal 0. This is a horizontal asymptote. 

In any scenario where the bottom of the fraction approaches ∞, the equation goes to 0. In any scenario where the top approaches ∞, the equation goes to ∞.

Lim x-->∞  y = 48/x2   is 0

Lim x-->∞ y = 48x2/x  = 48x   is ∞

Now that you have a simple understanding of what is happening, I will propose a general rule for the next examples to deal with every limit where x goes to infinity. If you try it for cases where x does not go to infinity, you will be totally screwed up, and I take no blame.

Lim x-->∞   y = (3x2 + 5x)/(7x3 – 2)

In any case where x is going to infinity involving a fraction, you must only care about the biggest factor of x in the equation. Every other term goes to 0. So this equation can be retyped as (0 + 0)/(7x3 – 0). Needless to say, this equation goes to 0. This is because the biggest factor is on the bottom.

Lim x-->   y = (5x3 – 2x2 + 17)/(8x2 – 4) = 5x3/0 = ∞ 

That was a case where the biggest factor was on top. What happens if there is a biggest factor on both the top and bottom?

Lim x-->   y = (x2 – 1)/(4x2 + 6x) = x2/4x2 = 1/4

That case has the biggest factor of x, x2, on both the top and bottom. But once we rewrite the equation, using my standard rule of largest factor of x, they will just cancel out, leaving behind their coefficients! In the following case there is a negative number in front of one of the biggest factors.

Lim x-->   y = (x2 – 1)/(-4x2 + 6x) = x2/-42 = -1/4

There are still more complex scenarios, including the ones where x goes to negative infinity, or the problem is set up so that the function (or y) will end up going to negative infinity.

In the following case it turns out that the equation goes to negative infinity. The reason is that using the previous methods, the only factor left is -5x3, and whenever the factor going to infinity is negative, it goes to negative infinity. (Note that if you are dealing with a fraction, and there is a remaining factor on the top and bottom with negative, they will cancel out, like -1's)

Lim x-->   y = (-5x3 – 2x2 + 17)/(8x2 – 4) = -5x3/0 = -∞

In the following case you must realize that negative infinity squared is positive, and cubed is negative again, like any negative number. Do not confuse it with the past case where there is a negative coefficient!

Lim x-->-  y = x2 = (-∞)2 =

Lim x-->-  y = x3 = (-∞)3 = -

 

One sided limits  

If I say lim x--> 0- then I mean if x is approaching 0 from the left, on a graph. If I say lim x --> 0+ then I mean x is approaching specifically from the right. This is important in some cases.

Lim x-->0+ y = ln x  If you know what the graph looks like then you know the answer is   -∞. The point here is that I had to say from the right because on the graph of ln x there is no way for x to be less than 0, so it can’t be approaching from there. It is clear from the picture, but you can try to solve ln –2 on a calculator; you will get an error. 

Lim x-->0- y = 3/x  = -∞. If I had said 0+ then the equation would be big on the positive side. Look at the graph 1/x, below. It is comparable to 3/x. It does approach negative infinity as the graph gets close to 0, and once you cross it, it comes down from positive infinity. 

 

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