Proof
(a) Prove that
(b) Prove that
(c) Let L be a real number. Prove that if
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- lim f(x) as x approaches infinity lim f(x) as x approaches negative infinityarrow_forward(Term-by-term Differentiability Theorem). Let fn be differentiable functions defined on an interval A, and assume ∞ n=1 fn(x) converges uniformly to a limit g(x) on A. If there exists a point x0 ∈ [a, b] where ∞ n=1 fn(x0) converges, then the series ∞ n=1 fn(x) converges uniformly to a differentiable function f(x) satisfying f(x) = g(x) on A. In other words, Proof. Apply the stronger form of the Differentiable Limit Theorem (Theorem6.3.3) to the partial sums sk = f1 + f2 + · · · + fk. Observe that Theorem 5.2.4 implies that sk = f1 + f2 + · · · + fk . In the vocabulary of infinite series, the Cauchy Criterion takes the followingform.arrow_forwardProof with limit definition that: limx→1/2 (1/x)=2 I have the following: Given ε>0. choose δ=? Suppose : 0<|x-(1/2)|<δ check: |(1/x)-2| from here I do not know how to get |x-(1/2)| from |(1/x)-2| in order to find δ?arrow_forward
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