![]() ![]() We now practice applying these limit laws to evaluate a limit. Root law for limits: lim x → a f ( x ) n = lim x → a f ( x ) n = L n lim x → a f ( x ) n = lim x → a f ( x ) n = L n for all L if n is odd and for L ≥ 0 L ≥ 0 if n is even and f ( x ) ≥ 0 f ( x ) ≥ 0. ![]() Power law for limits: lim x → a ( f ( x ) ) n = ( lim x → a f ( x ) ) n = L n lim x → a ( f ( x ) ) n = ( lim x → a f ( x ) ) n = L n for every positive integer n. Quotient law for limits: lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M for M ≠ 0 M ≠ 0 Product law for limits: lim x → a ( f ( x ) lim x → a f ( x ) = c L lim x → a c f ( x ) = c.Sum law for limits: lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) = L + M lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) = L + Mĭifference law for limits: lim x → a ( f ( x ) − g ( x ) ) = lim x → a f ( x ) − lim x → a g ( x ) = L − M lim x → a ( f ( x ) − g ( x ) ) = lim x → a f ( x ) − lim x → a g ( x ) = L − MĬonstant multiple law for limits: lim x → a c f ( x ) = c Then, each of the following statements holds: Assume that L and M are real numbers such that lim x → a f ( x ) = L lim x → a f ( x ) = L and lim x → a g ( x ) = M. Let f ( x ) f ( x ) and g ( x ) g ( x ) be defined for all x ≠ a x ≠ a over some open interval containing a. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions. The first two limit laws were stated in Two Important Limits and we repeat them here. These two results, together with the limit laws, serve as a foundation for calculating many limits. We begin by restating two useful limit results from the previous section. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. In this section, we establish laws for calculating limits and learn how to apply these laws. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. 2.3.6 Evaluate the limit of a function by using the squeeze theorem.2.3.5 Evaluate the limit of a function by factoring or by using conjugates.2.3.4 Use the limit laws to evaluate the limit of a polynomial or rational function.2.3.3 Evaluate the limit of a function by factoring.2.3.2 Use the limit laws to evaluate the limit of a function.But if you don't know the chain rule yet, this is fairly useful. But you could also do the quotient rule using the product and the chain rule that you might learn in the future. Now what you'll see in the future you might already know something called the chain rule, or you might You could try to simplify it, in fact, there's not an obvious way Plus, X squared X squared times sine of X. This is going to be equal to let's see, we're gonna get two X times cosine of X. Actually, let me write it like that just to make it a little bit clearer. So that's cosine of X and I'm going to square it. All of that over all of that over the denominator function squared. The derivative of cosine of X is negative sine X. Minus the numerator function which is just X squared. V of X is just cosine of X times cosine of X. ![]() So it's gonna be two X times the denominator function. So based on that F prime of X is going to be equal to the derivative of the numerator function that's two X, right over Of X with respect to X is equal to negative sine of X. So that is U of X and U prime of X would be equal to two X. Well what could be our U of X and what could be our V of X? Well, our U of X could be our X squared. So let's say that we have F of X is equal to X squared over cosine of X. We would then divide by the denominator function squared. Get if we took the derivative this was a plus sign. If this was U of X times V of X then this is what we would The denominator function times V prime of X. Its going to be equal to the derivative of the numerator function. Then the quotient rule tells us that F prime of X is going to be equal to and this is going to lookĪ little bit complicated but once we apply it, you'll hopefully get a little bit more comfortable with it. So for example if I have some function F of X and it can be expressed as the quotient of two expressions. But here, we'll learn about what it is and how and where to actually apply it. It using the product rule and we'll see it has some Going to do in this video is introduce ourselves to the quotient rule. ![]()
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