Chapter 9
and happens because the derivative of a small, rapidly oscillating function can be large. Example 9.6. De ne f n: R !R by f n(x) = x2 p x2 + 1=n: If x6= 0, then lim n!1 x 2 p x2 + 1=n = x jxj = jxj. 9.2. Uniform convergence 169 while f n(0) = 0 for all n2N, so f n!jxjpointwise on R. Moreover, f0 n (x) = x3 + 2x=n (x2 + 1=n)3=2! 8 >< >:
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