Answer to Question #242417 in Calculus for lalaland

Question #242417

Evaluate each of the following limits. Simplify all answers completely. Show all work toward your answer or provide an explanation for how you arrived at your answer.


1) lim x--> (x^2 -1)/(x^2 -8x+7)


2) lim x-->0^negative(-) (2)/(tan(x))


3) lim x--> infinity (cos(t))/(e^3t)


4) lim x--> (-)infinity (-3x^3 +5x^2 -4)




1
Expert's answer
2021-09-27T15:08:57-0400

1)


limx1x21x28x+7=limx1(x1)(x+1)(x1)(x7)\lim\limits_{x\to1}\dfrac{x^2-1}{x^2-8x+7}=\lim\limits_{x\to1}\dfrac{(x-1)(x+1)}{(x-1)(x-7)}

=limx1x+1x7=1+117=13=\lim\limits_{x\to1}\dfrac{x+1}{x-7}=\dfrac{1+1}{1-7}=-\dfrac{1}{3}

2)


limx02tanx=\lim\limits_{x\to0^-}\dfrac{2}{\tan x}=-\infin

=limx1x+1x7=1+117=13=\lim\limits_{x\to1}\dfrac{x+1}{x-7}=\dfrac{1+1}{1-7}=-\dfrac{1}{3}



3)


limtcostet=0\lim\limits_{t\to\infin}\dfrac{\cos t}{e^t}=0

4)


limx(3x3+5x24)=\lim\limits_{x\to-\infin}(-3x^3+5x^2-4)=\infin


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