Chapter4(0211)

Chapter4(0211) - Ch4/MATH0211/YMC/2009-10 1 Chapter 4....

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Unformatted text preview: Ch4/MATH0211/YMC/2009-10 1 Chapter 4. Limits and Continuity 4.1. Limits The concept of a limit lies at the foundation of calculus. The idea involves the notion of getting closer and closer to something, but yet not touching it. Limits are used to define continuity and deriva- tives. Let f ( x ) be a function and a R . If f ( x ) can be made to be as close to L as desired by making x sufficiently close to a , we say that the limit of f ( x ) as x approaches a is L , and we write lim x a f ( x ) = L. Example 4.1 Consider the function f ( x ) = x 2- 1 x- 1 . How does the function behave near x = 1 ? Solution . Observe from the defining formula of f that f ( x ) is defined for any real number except x = 1 . For x 6 = 1 , we can simplify the function: f ( x ) = ( x- 1)( x + 1) x- 1 = x + 1 for x 6 = 1 . The graph of f is therefore the straight line y = x + 1 with the point (1 , 2) removed. Note that even though the function is not defined at x = 1 , the value of f ( x ) gets closer and closer to 2 as x gets closer and closer to 1 . f (0 . 9) f (0 . 99) f (0 . 999) f (1 . 001) f (1 . 01) f (1 . 1) 1 . 9 1 . 99 1 . 999 2 . 001 2 . 01 2 . 1 Therefore we can write lim x 1 f ( x ) = 2 , or lim x 1 x 2- 1 x- 1 = 2 . Ch4/MATH0211/YMC/2009-10 2 We emphasize that, when computing a limit, we are NOT concerned with what happens to f ( x ) when x = a , but only with what happens to f ( x ) when x is sufficiently close to a . In fact, the limit value does not depend on how the function is defined at a . Example 4.2 The function f in the following figure has limit 2 as x 1 even though f is not defined at x = 1 (as in Example 4.1). The function g has limit 2 as x 1 even though g (1) 6 = 2 . The function h is the only one whose limit as x 1 equals its value at x = 1 , that is, we have lim x 1 h ( x ) = h (1) . This is a special situation and we shall return to this when we discuss continuity. Example 4.3 1. If f ( x ) = c is a constant function, then lim x a f ( x ) = lim x a c = c. For instance, lim x 1 9 = 9 and lim x - 2 9 = 9 . 2. If f ( x ) = x , then lim x a f ( x ) = lim x a x = a. For instance, lim x 5 x = 5 . 2 Properties of Limits If lim x a f ( x ) and lim x a g ( x ) exist, then 1. lim x a [ f ( x ) g ( x )] = lim x a f ( x ) lim x a g ( x ) That is, the limit of a sum or difference is respectively the sum or difference of the limits. 2. lim x a [ f ( x ) g ( x )] = lim x a f ( x ) lim x a g ( x ) That is, the limit of a product is the product of the limits. Ch4/MATH0211/YMC/2009-10 3 3. lim x a [ kf ( x )] = k lim x a f ( x ) That is, the limit of a constant times a function is the constant times the limit of the function....
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Chapter4(0211) - Ch4/MATH0211/YMC/2009-10 1 Chapter 4....

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