Answer to Question #212381 in Calculus for Kenneth

Question #212381

1. Evaluate the following limits

(a) Lim y"\\to" -1 Fourth root of 4y^3 minus 3

(b) lim x"\\to" infinity 1+x all over 1-x

(c) lim x "\\to" 2 x - 2 all over 4 - x^2

(d) lim x "\\to" 0 sinx - x cos x all over x^3


1
Expert's answer
2021-07-01T13:34:11-0400

(a)


"\\lim\\limits_{y\\to-1}(\\sqrt[4]{y^3}-3)=DNE\\ (\\text{does not exist)}"

since we consider "y\\in \\R, y\\geq0."


(b)


"\\lim\\limits_{x\\to\\infin}(\\dfrac{1+x}{1-x})=\\lim\\limits_{x\\to\\infin}\\bigg(\\dfrac{\\dfrac{1}{x}+\\dfrac{x}{x}}{\\dfrac{1}{x}-\\dfrac{x}{x}}\\bigg)"

"=\\lim\\limits_{x\\to\\infin}\\bigg(\\dfrac{\\dfrac{1}{x}+1}{\\dfrac{1}{x}-1}\\bigg)=\\dfrac{0+1}{0-1}=-1"

(c)


"\\lim\\limits_{x\\to2}(\\dfrac{x-2}{4-x^2})=\\lim\\limits_{x\\to2}\\bigg(\\dfrac{x-2}{(2-x)(2+x)}\\bigg)"

"=\\lim\\limits_{x\\to2}\\big(\\dfrac{-1}{2+x}\\big)=-\\dfrac{1}{2+2}=-\\dfrac{1}{4}"

(d)


"\\lim\\limits_{x\\to0}(\\dfrac{\\sin x-x\\cos x}{x^3})"

"\\lim\\limits_{x\\to0}(\\sin x-x\\cos x)=\\sin(0)-0(\\cos(0))=0"

"\\lim\\limits_{x\\to0}(x^3)=(0)^3=0"

"\\dfrac{0}{0}=>\\lim\\limits_{x\\to0}(\\dfrac{\\sin x-x\\cos x}{x^3})=\\lim\\limits_{x\\to0}\\dfrac{(\\sin x-x\\cos x)'}{(x^3)'}"

"=\\lim\\limits_{x\\to0}\\dfrac{\\cos x-\\cos x+x\\sin x}{3x^2}=\\lim\\limits_{x\\to0}\\dfrac{\\sin x}{3x}"

"=\\dfrac{1}{3}\\lim\\limits_{x\\to0}\\dfrac{\\sin x}{x}=\\dfrac{1}{3}(1)=\\dfrac{1}{3}"

We use that "\\lim\\limits_{x\\to0}\\dfrac{\\sin x}{x}=1"



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