notes - x →-2 x 2 x-4 x 2 = lim x →-2 x-4 =-6 Next you...

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On the assigned problems from 2.5, #1 is similar to problem #43 on page 134. Each part of this problem is similar, so I’ll only do part (a). #43, pg 143(a) Show that f ( x ) = x 2 - 2 x - 8 x + 2 has a removable discontinuity at x = - 2 and ﬁnd a function which agrees with f for all x 6 = - 2 and is continuous on R . First you need to know what a removable discontinuity is. A function h ( x ) has a removable discontinuity at x = a when lim x a h ( x ) = L and f ( a ) 6 = L . In words, this means a function has a removable discontinuity when the limit exists but is not equal to the function value. So to show that the given f ( x ) has a removable discontinuity at x = - 2, compute the limit lim x →- 2 x 2 - 2 x - 8 x + 2 = lim
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Unformatted text preview: x →-2 ( x + 2)( x-4) x + 2 = lim x →-2 ( x-4) =-6 . Next you try to compute f (-2) = , so f (-2) is undeﬁned. In partic-ular, this means the limit of the function and the value of the function are not equal. So f ( x ) has a removable discontinuity at x =-2. Now I want a continuous function g ( x ) which agrees with f every-where except at x =-2. I’ll do this with a piecewise function. Let g ( x ) = x 2-2 x-8 x + 2 x 6 = 2 x-4 x = 2 . Now you can check an see that g ( x ) is continuous everywhere and agrees with f ( x ) except at x =-2. 1...
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