 While we can try to find limits numerically, we usually want to find these limits algebraically. In order to find limits algebraically, we might begin with an important limit, the limit as x approaches a of x. We'll find this limit numerically by picking x values close to a, and then finding out what x is equal to. So if x is a minus 0.1, then x is a minus 0.1. And we can find our values of x for these other values of x. And so it appears that as x approaches a, the limit is also going to be a. Now it might not be obvious why this is such a big deal, but let's consider the problem of finding the limit as x approaches 2 of x squared plus 4x plus 7. Since our expression is a sum, then we can use the theorem regarding the limit of a sum and break it apart as the limit as x approaches 2 of x squared plus the limit of 4x plus the limit of 7. Now our first term is the limit of x squared. And since this is the product of x times x, we can evaluate it as the limit of x times the limit of x. Our next term is 4x, which is a constant multiple, so we can remove that factor of 4 to the front of the limit expression. Now we have the limit as x approaches 2 of x appearing 3 times in our expression. And we've already calculated what that is, so we can substitute that in. We also have this limit as x approaches 2 of 7. Since 7 is a constant, then the limit is just going to be 7 itself. So we'll include that, and we'll evaluate our expression to get the limit of 19. While the actual value of the limit is 19, if we look closely as to how we calculated it, we might see that we found 2 to the second plus 4 times 2 plus 7. And that's exactly the same thing we would have done if we just evaluated the polynomial x squared plus 4x plus 7. Since a polynomial is a sum of terms that might include constants or products of variables multiplied by constant terms, this suggests the following theorem. Let f of x be a polynomial, then the limit as x approaches a of f of x is f of a itself. What about a rational function like x squared plus 3x plus 7 over x cubed plus 9x plus 4? Since this is a quotient, we can find the quotient of the limits. But now we're looking at the limits of 2 polynomials. And so we can evaluate both of these limits by evaluating the polynomials at the limit point 3. And this suggests the following. Let f of x be a rational function, then the limit as x approaches a of f of x is f of a provided that our rational function is actually defined at x equals a. How about a radical function? Suppose f of x is a polynomial, let's find the limit as x approaches a of the square root of f of x. We'll proceed a little bit differently to introduce a key mathematical concept. Even if we don't know what something is, we can introduce a variable so that we can at least talk about it. In this case, suppose our limit is equal to l. Then if I multiply the limit by itself, I get the limit of the square root of f of x times the square root of f of x. On the one side, I get l squared. On the other side, provided that f of x corresponds to a positive number, and it has to because we're going to be taking the square root of it, I get the limit as x approaches a of f of x. And that tells me that l is equal to the square root of the limit as x approaches a of f of x. And this suggests another important limit theorem. Suppose f is a polynomial or a rational function, then the limit as x approaches a of the nth root of f of x is the nth root of the limit, provided that the latter exists. And if we put these things together, it tells us that we can find the limit of a polynomial by evaluating it at the limiting point. We can also find the limit of a rational function by evaluating it at the limiting point. We can also find the limit of a root function by evaluating it at the limiting point. Now, any function that we can find by taking a polynomial and forming a quotient or a root of that polynomial is called an algebraic function. And so these theorems together say that if I have an algebraic function, if it's a function that can be expressed in terms of powers, roots, and or quotients of the variable x, our limit is just going to be the function value provided that the function is actually defined at that point. So if I want to find the limit as x approaches 3 of the square root of 100 minus 4x minus x squared, this is an algebraic function, and so I can evaluate it by using our limit theorems. Since it's the limit of a square root, I can find this by taking the square root of a limit. Since this is a limit of a polynomial, I can evaluate it by evaluating the polynomial, and that allows me to calculate the limit. Now remember when we started talking about limits, we made a big deal about the fact that the value of the function is irrelevant to the value of the limit. And yeah, we just went through a lot of work to show that the limit of an algebraic function is the value of the function at the limit provided that the value exists. And so this leads to the obvious question, to find the limit as x approaches a of f of x, why don't we just evaluate f of a? And sometimes we can. The limit as x goes to 4 of f of x equals 3x is 3 times 4 or 12. And here's a case where we can just find the limit by evaluating the function. But in other cases, we can't. If I want to find the limit as x goes to 1 as square root of x minus 1 over x minus 1, I find that at x equal to 1, the numerator and denominator are both 0, and 0 over 0 is undefined. And so we can only find the limit by evaluating the function if the function actually is defined at that point. So what do we do in that case? Before moving on, it'll be helpful to introduce three terms. Given an arithmetic expression, that expression is either determinant, it can be evaluated as it's written. For example, 3 squared or 5 plus the cube root of 3 divided by the 7 plus square root of 11. The expression might also be undefined. It cannot be evaluated because it violates a primary rule of mathematics. For example, the expression 5 over 0 violates the rule, the denominator cannot be 0. And finally, the most important case, the expression might be indeterminate. It's undefined, but another rule suggests an answer. For example, the expression 0 over 0 violates the rule, the denominator can't be 0. However, the rule any fraction with numerator 0 is 0 suggests that an answer is possible. And here's a useful idea to remember. If f of a has an indeterminate form, then the limit as x approaches a might be found using some sort of algebraic simplification. We'll take a look at those simplifications next.