 Recall that a function is continuous if the function evaluation agrees with the limit calculation. And so what we care about in this video is how do we can we combine continuous functions together? So if F and G are continuous functions at the point X equals A, then when we do the four arithmetic operations to these functions, combine them in that way, that'll preserve the continuity. That is to say, if F and G are continuous at A, F plus G will be continuous at A. So will F minus G and so will F times G. The one put potential exceptions when you do division, F divided by G will be continuous so long as G of A is not zero. If the dom or goes to zero, then continuity is not guaranteed. In fact, we wouldn't expect it because that would imply that the function is undefined at that, at the location. Let me give you an argument for addition. The proofs for subtraction, division, and multiplication will follow similarly because these actually follow very quickly from limit properties. So if we do the additive statement, right? So since F is continuous, that means the limit as X approaches A of F of X is equal to F of A. And because G is continuous, that means the limit as X approaches A of G of X is gonna be G of A. That's what we're talking about. The function agrees with the limit. Now let's consider the limit of F plus G as X approaches A. Well, when you have the expression F plus G of X, what is the sum of two functions due to the number X? Well, it's going to assign to X the numbers F plus, or F of X and G of X, but we add them together by addition, all right? So that's just the definition of function addition. Then by limit properties, if you take the limit of a sum, this becomes a sum of two limits for which then the limit of F of X plus G of X will become the limit of F of X plus the limit of G of X. Now the limit of F of X because F is continuous will just be F of A. The limit of G of X because G is continuous will be G of A, in which case F of A plus G of A is exactly what F plus G at A does. And so we see here that the limit of F plus G as X gets close to A is just F plus G evaluated at A. So this proves that F plus G is continuous and a similar property of limits, particularly this equality right here can be used to modify this proof to show division, multiplication, and subtraction. So let's see how we can use these properties. Let's suppose we wanna find the limit as X approaches negative two of X cubed plus two X squared minus one over five minus three X. Because we have a limit of a quotient, we're gonna be looking at the quotient of limits. So we're gonna take the limit as X approaches negative two of X cubed plus two X squared minus one over the limit as X approaches negative two of five minus three X. So long as the denominator doesn't go to zero, this is a valid thing to do with our limit calculation. And these are just polynomials because they're continuous, I can compute their limits by direct substitution. For which let's take a look at the denominator first. If I plugged in negative two, we're gonna get five minus three times negative two for which I'm just checking out the denominator so that I could know that the denominator doesn't go to zero because if it goes to zero, I'll have to treat this one differently. Negative three times negative two is a positive six. Notice that five plus six, that's gonna be positive 11. So we know the denominator doesn't go to zero. Let's take a look at the numerator. If we plug in negative two, we're gonna get negative two cubed plus two times negative two squared minus one for which case negative two cubed is gonna be a negative eight. Negative two squared is a positive four times that by two is a positive eight. You get a minus one. You'll see that the negative eight cancels with positive eight and you end up with negative one over 11. So negative one 11th would be the limit here. So because the function was continuous, we are able to simply just plug in the value negative two to compute the limit. And this was guaranteed because combinations of addition, subtraction, multiplication, division will preserve the continuity of these functions. A rational function is continuous on its domain because it's a quotient of polynomials that are continuous on their domains. Let's take a look at the example where we wanna find the limit as x approaches pi of sine of x over two plus cosine of x. Well, like we saw in the previous example, we're gonna wanna consider the limit of sine of x as x goes to pi divided by the limit as x approaches pi of two plus cosine of x, right? As long as the denominator doesn't go to zero, we can just plug in pi to find the limit because sine is a continuous function. All the trig functions are continuous on their domains. And two plus, so cosine is continuous, a constant function two is continuous, added together will be continuous. The quotient will be continuous as long as the denominator doesn't go to zero. So let's think about the denominator for a moment. We're gonna get two plus cosine of pi. The numerator is gonna be sine of pi for which then continuing on here, cosine of pi is actually a negative one. So we're gonna get two minus one. That's definitely not zero. The denominator is gonna be a one. On the other hand, sine of pi is zero. It's perfectly fine if the numerator is zero. It's just we don't want the denominator to go to zero because that would give us some type of discontinuity. If the numerator of a fraction's zero, that is a rational number. That's just gonna be zero. And so we see that the limit here is just gonna be zero. We can use the fact that because we've combined together continuous functions in an algebraic sense, that is using addition, subtraction, multiplication, division, then we can compute the limits of these things just by direct substitution. All right, what about the function f of x equals the natural log of x plus arc tangent of x divided by x squared minus one? Where is that function continuous? There's a lot going on there. So notice the numerator. The numerator is the natural log plus arc tangent. So the function natural log plus arc tangent will be continuous so long as natural log is continuous. And where arc tangent is continuous, we'll just take the intersection there. And so the numerator is continuous so long as the natural log and arc tangent are continuous. But what about the denominator? Well, x squared minus one, this is a polynomial. It'll be continuous and thus the ratio will be continuous so long as the denominator doesn't go to zero and the numerator is continuous. But the natural log we've learned previously, all logarithms are continuous on their domains. Arc tangent is an inverse trigonometric function. It'll be continuous on its domain as well. So because adding, subtracting, multiplying, dividing continuous functions preserves continuity, asking where is the function continuous is basically just asking in this context what's the domain of the function? The two questions are basically saying the same thing because of the properties we saw earlier, this function will be continuous on its domain. So what's its domain? Well, it depends on which piece you're looking at. For the natural log, we need that x is greater than zero. If x is greater than zero, the natural log, that's its domain, it'll be continuous right there. In terms of arc tangent, arc tangent, its domain is all real numbers. So there's no concern with that whatsoever. How about x squared minus one? Well, as a polynomial, its domain is all real numbers, no restriction right there. So the other concern, of course, is the division. What makes the denominator go to zero? And so then we have to investigate the equation x squared minus one equals zero. But notice, if x squared minus one equals zero, that means x squared equals one, take the square root, we get x equals plus or minus the square root of one, that is x equals plus or minus one. These are values we should avoid. So what we see here is that the domain of this function, if you look at the first piece, you need to be from zero to infinity. That's what the natural log says. For the division part, we have negative infinity to negative one union, negative one to one, union one to infinity. And so when we put these things together, we see that the domain of our function is going to be zero to one, union one to infinity. And therefore, since this function's continuous on its domain, it'll be continuous from zero to one and one to infinity.