微积分第二章课件 Limits and Derivatives.ppt

Example 3 Prove that Solution 1. Guessing a value for δ. Let ε be a given positive number. We want to find a number δ such that that is, This suggests that we should choose 2. Showing that this δ works. Given ε0, let If , then This shows that Example 4 Prove that Solution 1. Guessing a value for δ. Let ε0 be given. We have to find a number δ0 such that that is, Notice that if we can find a positive constant C such that then and we can make by taking We can find such a number C if we restrict x to lie in some interval centered at 3. In fact, since we are interested only in values of x that are close to 3, it is reasonable to assume that x is within a distance 1 from 3, that is, Then so Thus, we have and so C = 7 is a suitable choice for the constant. But now there are two restrictions on namely To make sure that both of these inequalities are satisfied, we take δ to be the smaller of the two numbers 1 and ε/7. The notation for this is 2. Showing that this δ works. Given ε0, let If , then We also have so This shows that Infinite Limits Definition 4 Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then means that for every positive number M there is a positive number δ such that Example 5 Prove that Solution 1. Guessing a value for δ. Given M 0, we want to find δ0 such that That is, This suggests that we should take 2. Showing that this δ works. If M 0 is given, let If , then Therefore, by Definition 4, 2.5 Continuity Definition 1 A function f is continuous at a number a if The kind of discontinuity illustrated in part (3) of Figure 1 is called a removable discontinuity because we could remove the discontinuity by redefining at c. The discontinuity