Concept explainers
Proof
(a) Given that
prove that there exists an open interval (a, b) containing 0 such that
(b) Given that
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Calculus of a Single Variable
- (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_forwardlim x-->3 f(x)=8 and lim x--->3 g(x)= -4 1) limx--->3 (3f(x)-5g(x)) 2) lim x---> G(x)^3 + 50 / ( 5f(x)^(2/3) )arrow_forwardA. Does f(1) exist B. Does lim x->1 f(x) exist C. Does lim x->1 f(x) equal f(1) D. Is the function continuous at x=1arrow_forward
- lim x to infinity x-2/x2+1 find the limitarrow_forward(Right and Left Limits). Introductory calculus coursestypically refer to the right-hand limit of a function as the limit obtained by“letting x approach a from the right-hand side.” (a) Give a proper definition in the style of Definition 4.2.1 ((Functional Limit).for the right-hand and left-hand limit statements: limx→a+f(x) = L and limx→a−f(x) = M. (b) Prove that limx→a f(x) = L if and only if both the right and left-handlimits equal L.arrow_forwardlim x→∞arrow_forward
- 1- let f(x)= |x^2-4|/(x+2). Determine the limit lim_{x->-2} f(x), if it exist. 2- determine the limit lim_{h->0}[(x+h)2-x^2]/h by treating x as a constant. 3- dtermine the lim_{x->0} cos(2x)/x 4-determine the limit_{x->0} sin(2x^2)/xtan(4x)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_forwardlim x → 0− g(x) lim x → 0+ g(x) Is g continuous at x = 0?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning