Let lim 0 Given 0 \u210e 0 such that fx L when 0x c Let 2 fx L 2 when x 2 fx L 2

# Let lim 0 given 0 ℎ 0 such that fx l when 0x c let

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Let lim 𝑥→𝑐 ?(𝑥) = ? > 0 . Given 𝜖 > 0, ?ℎ??? ?𝑥𝑖??? 𝛿 > 0 such that |f(x)-L|< 𝜖 when 0<|x-c|< 𝛿 . Let 𝜖 = 𝐿 2 |f(x)-L|< 𝐿 2 when x ∈ ? ∩ 𝐷 - 𝐿 2 < f(x)-L< 𝐿 2 when x ∈ ? ∩ 𝐷 L- 𝐿 2 < f(x) < L+ 𝐿 2 when x ∈ ? ∩ 𝐷 f(x)>0 when x ∈ ? ∩ 𝐷 because L- 𝐿 2 >0 18) Let lim(f(x))=L when L . Suppose 𝜖 = 2 , Then there exists 𝛿 > 0 such that |f(x)-L<2 and 0<|x-c|< 𝛿 where x ∈ 𝐷 . |f(x)-L|<2 when x ∈ ? ∩ 𝐷 |f(x)|=|f(x)-L+L| |f(x)-L|+|L|< 2+|L| for any x ∈ ? ∩ 𝐷. Let M 1 =2+|L|, then |f(x)| M 1 for any x ∈ ? ∩ 𝐷. and M = max{M 1 , |f(c)|}. Extra Problems 1a) S n converges to 0 1b) S n doesn t converge; Subsequential lim S={0, 1, +∞} ; lim Inf S n =0; lim Sup S n = +∞ 1c) S n doesn t converge; Subsequential lim S={0, +∞} ; lim Inf S n =0; lim Sup S n = +∞ 1d) S n does not converge; Subsequential lim S={1, √3 2 , 1 2 , 0, − 1 2 , − √3 2 , −1} ; lim Inf S n =-1; lim Sup S n = 1 2) V. b, c, and d are true using theorem 5.1.1 3) Given 𝜖 > 0 𝑎?? 𝛿 > 0 , |f(x)-L|< 𝜖 such that 0<|x-c|< 𝛿 . Let 𝛿 = 𝜖 3 |(3x+4)-10|< 𝜖 |3x-6|< 𝜖 3|x-2|< 𝜖 for all x

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• Fall '08
• Staff
• Limits, Limit of a sequence, Limit superior and limit inferior, lim sup, subsequence, Lim Sup Vn