Answer to Question #213707 in Calculus for amit

Question #213707

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
2021-07-27T17:05:08-0400

(a)


limy1(4y343)=DNE,\lim\limits_{y\to -1}(\sqrt[4]{4y^3}-3)=DNE,

does not exist.



limy1(4y343)=23\lim\limits_{y\to 1}(\sqrt[4]{4y^3}-3)=\sqrt{2}-3


(b)


limx1+x1x=limx1+x1x=limx1x+xx1xxx\lim\limits_{x\to \infin}\dfrac{1+x}{1-x}=\lim\limits_{x\to \infin}\dfrac{1+x}{1-x}=\lim\limits_{x\to \infin}\dfrac{\dfrac{1}{x}+\dfrac{x}{x}}{\dfrac{1}{x}-\dfrac{x}{x}}

=limx1x+11x1=0+101=1=\lim\limits_{x\to \infin}\dfrac{\dfrac{1}{x}+1}{\dfrac{1}{x}-1}=\dfrac{0+1}{0-1}=-1

(c)


limx2x24x2=limx2x2(2x)(2+x)=limx212+x\lim\limits_{x\to 2}\dfrac{x-2}{4-x^2}=\lim\limits_{x\to 2}\dfrac{x-2}{(2-x)(2+x)}=-\lim\limits_{x\to 2}\dfrac{1}{2+x}

=12+2=14=-\dfrac{1}{2+2}=-\dfrac{1}{4}



(d)


limx0(sinxxcosx)=00=0\lim\limits_{x\to 0}(\sin x-x\cos x)=0-0=0

limx0(x3)=0\lim\limits_{x\to 0}(x^3)=0

[00]\big[\dfrac{0}{0}\big]

L'Hospital's Rule

limx0sinxxcosxx3=limx0(sinxxcosx)(x3)\lim\limits_{x\to 0}\dfrac{\sin x-x\cos x}{x^3}=\lim\limits_{x\to 0}\dfrac{(\sin x-x\cos x)'}{(x^3)'}

=limx0cosxcosx+xsinx3x2=limx0sinx3x=\lim\limits_{x\to 0}\dfrac{\cos x-\cos x+x\sin x}{3x^2}=\lim\limits_{x\to 0}\dfrac{\sin x}{3x}

=limx0(sinx)(3x)=limx0cosx3=13=\lim\limits_{x\to 0}\dfrac{(\sin x)'}{(3x)'}=\lim\limits_{x\to 0}\dfrac{\cos x}{3}=\dfrac{1}{3}




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