Answer to Question #203189 in Real Analysis for Rajkumar

Question #203189

Show that

i) lim x→∞ [ (x-3) / (x-1) ] x = 1/e2


ii) lim x→5/3 1/ (3x+5)2 = ∞

1
Expert's answer
2021-06-09T17:01:56-0400

i)


limt0[1+t]1t=e\lim\limits_{t\to 0}[1+t]^{\tfrac{1}{t}}=e


limx[x3x1]x=limx[12x1]2(x12)+1\lim\limits_{x\to \infin}[\dfrac{x-3}{x-1}]^x=\lim\limits_{x\to \infin}[1-\dfrac{2}{x-1}]^{-2(-\tfrac{x-1}{2})+1}

=1(limx[12x1](x12))2=e2=1e2=1\cdot\bigg(\lim\limits_{x\to \infin}[1-\dfrac{2}{x-1}]^{(-\tfrac{x-1}{2})}\bigg)^{-2}=e^{-2}=\dfrac{1}{e^2}

ii)


limx5/31(3x+5)2=1(3(53)+5)2=1100\lim\limits_{x\to 5/3}\dfrac{1}{(3x+5)^2}=\dfrac{1}{(3(\dfrac{5}{3})+5)^2}=\dfrac{1}{100}

limx5/3[1(3x+5)2]=19limx5/3[1(x+53)2]=\lim\limits_{x\to -5/3}\big[\dfrac{1}{(3x+5)^2}\big]=\dfrac{1}{9}\cdot\lim\limits_{x\to -5/3}\bigg[\dfrac{1}{(x+\dfrac{5}{3})^2}\bigg]=\infin


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