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Unformatted text preview: 1.3 Calculating Limits In the last section we developed an intuitive understanding of what lim x a f ( x ) , lim x a + f ( x ) , and lim x a f ( x ) all mean. In particular, if we know what the graph of f looks like then we know how to find the values of each of these limits. These limits can be very formally defined but the definition is a little hard to fully understand and is beyond the scope of this course. In this section we will discuss how to find the values of these limits when the function f is given by a formula and we may not know, or may not want to have to determine, what its graph looks like. Limit Laws We start with what are known as limit laws . These are laws that can be rigorously proven from the definition of a limit and can be used to evaluate limits. Since we havent talked about the rigorous definition of the limit we cannot prove these laws. However, I hope that each law seems intuitively reasonable based on our intuitive understanding of what limits are. Law 1: lim x c x = lim x c + x = lim x c x = c Law 2: If k is a constant then lim x c k = lim x c + k = lim x c k = k 1 Laws 3  6 below can only be applied if lim x c f ( x ) and lim x c g ( x ) both exist. Law 3: lim x c ( f ( x ) g ( x )) = lim x c f ( x ) lim x c g ( x ). Law 4: lim x c f ( x ) g ( x ) = lim x c f ( x ) lim x c g ( x ) . Law 5: lim x c f ( x ) g ( x ) = lim x c f ( x ) lim x c g ( x ) provided lim x c g ( x ) 6 = 0. Law 6: lim x c f ( x ) r = lim x c f ( x ) r provided f ( x ) is positive in a neighborhood of c (except possibly at c itself) in the case when r = p/q and q is even. Note: These laws also apply when lim x c is replaced everywhere either by lim x c or by lim x c + . For example, replacing lim x c by lim x c + in law 3 we see that if lim x c + f ( x ) and lim x c + g ( x ) are both defined then lim x c + ( f ( x ) g ( x )) = lim x c + f ( x ) lim x c + g ( x ). Examples: 1. Use the limit laws to evaluate lim x 2 (2 x 3 5 x + 2). 2. Use the limit laws to evaluate lim x 9 x + 7 x 2 21 . Building up from Limit Laws Polynomials: If p ( x ) is a polynomial and c is any real number then lim x c + p ( x ) = lim x c p ( x ) = lim x c p ( x ) = p ( c ) In other words, if the function is a polynomial then finding the limit of the function as x goes to c is the same as finding the value of the function at x = c . This is not surprising when you think about what graphs of polynomials look like. Rational functions: If f ( x ) = p ( x ) /q ( x ) is a rational function then p and q are polynomials so lim x c + p ( x ) = lim x c p ( x ) = lim x c p ( x ) = p ( c ) , and lim x c + q ( x ) = lim x c q ( x ) = lim x c q ( x ) = q ( c ) ....
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 Spring '07
 Vorel
 Calculus, Limits

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