Question

1. For each of the following use Theorem 3.3.4 to determine if the limit exists and the value of the limit when it does exist. (d) lim 1-40 |+2 (b) lim VH (e) lim / +2 ( limsin Theorem 3.3.4. Suppose f is defined in a deleted neighborhood of a point c. Then lim-f(x) exists and equals Lif and only if both lim + f(x) and lim- f(x) exist and have the same value L Proof. Suppose first that lim-f(x) exists and equals . Then for any € > 0, there exists a 8 >0 such that f(x)-[<e for 0 < x-c<3, that is, (x) < for both -8 <x < < 0 and 0<x-c<8. Thus we have lim- f(x)=L, and lim f (x)=L Conversely, suppose both the left-hand and the right hand limits exist and equal the common value L. Then for any e > 0, there exists a dj > Osuch that Tx)-4<e for- & <x-c<0 and there exists a & > O such that f(x)-4<c for 0<r-c Thus, if 0 < - < -Min (8.82), we have f(x) - <e, and it follows that limef(x)=L

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( real analysis )

Answer 1

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