Let f : D→R be a function and let c be a limit point of D. Assume?

Assume f(x)→L as x→c：


(a) Prove that |f(x)| → |L| as x → c


(b) If f(x)%26gt;= 0 for all x ∈ D, prove that square root of f(x) →square root of L as x → c

Let f : D→R be a function and let c be a limit point of D. Assume?
(a) Fix ε%26gt;0. Since f(x)→L as x→c, we know that there is a δ%26gt;0 such that if |x-c|%26lt;δ then |f(x)-L|%26lt;ε. But note that by the triangle inequality, | |f(x)| - |L| | ≤ |f(x) - L|, and so the same δ works.





[Triangle Inequality revisited: Notice that for any real A and B,


|A| = |A-B + B|


≤ |A-B| + |B|, so


|A| - |B| ≤ |A-B|. But likewise,


|B| = |B-A + A|, and so by the same argument,


|B| - |A| ≤ |B-A| = |A-B|. Therefore


| |A| - |B| | ≤ |A-B|. ]





(b) Fix ε%26gt;0. By a similar argument as above, just use the δ associated with ε².





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