Answer to Question #285799 in Real Analysis for Reens

Question #285799

show that


a) lim x- infinity (x-3/x-1)^x =1/e^2


b) lim x->5/3 1/(3x+5)^2 =infinity

1
Expert's answer
2022-01-10T14:52:45-0500

a)


limx(x3x1)x=limx(12x1)x\lim\limits_{x\to\infin}\bigg(\dfrac{x-3}{x-1}\bigg)^x=\lim\limits_{x\to\infin}\bigg(1-\dfrac{2}{x-1}\bigg)^x

=limx(12x1)x12(2x1)x=\lim\limits_{x\to\infin}\bigg(1-\dfrac{2}{x-1}\bigg)^{-{x-1 \over 2}(-{2 \over x-1})x}

=limx[(12x1)x12]2xx1=\lim\limits_{x\to\infin}\bigg[\bigg(1-\dfrac{2}{x-1}\bigg)^{-{x-1 \over 2}}\bigg]^{-{2x \over x-1}}

=e2=1e2=e^{-2}=\dfrac{1}{e^2}

b)


limx5/3(13x+5)2=(13(5/3)+5)2=1100\lim\limits_{x\to5/3}\big(\dfrac{1}{3x+5}\big)^2=\big(\dfrac{1}{3(5/3)+5}\big)^2=\dfrac{1}{100}

limx5/3(13x+5)2=19limx5/3(1x+5/3)2=\lim\limits_{x\to-5/3}\big(\dfrac{1}{3x+5}\big)^2=\dfrac{1}{9}\lim\limits_{x\to-5/3}\big(\dfrac{1}{x+5/3}\big)^2=\infin


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