 A rational function is a quotient of two polynomials. Since rational functions don't involve any operations besides those of arithmetic, they are algebraic functions, so we know that, as long as it's defined at x equals a, the limit as x approaches a is the function value itself. This will be useful, but we may need to do a little algebra to get there. So let's find this limit. Now maybe we're lucky and the function is defined at x equals 3. We find, unfortunately, it isn't defined at x equals 3 because there's a 0 in the denominator. However, it has indeterminate form 0 over 0, so an algebraic simplification may lead to an answer. Now there's a couple ways of doing this, but a useful theorem can be very helpful. If f of x is a polynomial where f of a is 0, then f of x has a factor of x minus a. Since x equals 3 made both numerator and denominator equal to 0, then both numerator and denominator have a factor of x minus 3. And the significance of this is factoring is very hard, so it helps when you know what one of the factors is. So we know that our numerator is x minus 3 times something, and the only something that could possibly be a factor is x plus 5. Similarly, the denominator is x minus 3 times something, and again, the only thing that something could be is x minus 5. And so we could remove the common factors. Now the new rational function is algebraic and defined at x equals 3, so the limit is the function value at x equals 3, which will be, and we can find this limit. And again, we find that the numerator and denominator are both 0 at x equals negative 3. Then x plus 3 is a factor of both. So we find our numerator is x plus 3 times, and our denominator is x plus 3 times, and so we'll remove the common factor. And we have an algebraic function that's defined at negative 3, so the limit will be the function value, which is, and we can even do this for horrifying expressions like this. Since x equals 1 makes both numerator and denominator equal to 0, both have a factor of x minus 1, and we can find the other factor by division. So pick your favorite method, synthetic division, and we find. And so our numerator and denominator factor as, and we can remove the common factor. And because we're dealing with the rational expression, we can only have equality as long as x is not equal to 1, but because we're taking a look at the limit, we don't actually care about what happens at x equals 1. So we can replace one expression with the other. And at x equals 1, the numerator and denominator of our new expression are, again, both equal to 0. But that means that both have a factor of x minus 1, and dividing gives us the other factor. And so we find, and so our new rational expression can be simplified to for x not equal to 1, but this time we still don't care about what happens at x equals 1, so we can replace the old algebraic expression with the new one. And in this expression, at x equals 1, our numerator and denominator are not both equal to 0. Actually, that's a good thing since the algebraic function is defined, and so the limit is equal to the function value, which will be...