Answer to Question #242417 in Calculus for lalaland

Question

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)

Answer 1

1)

$$\lim\limits_{x\to1}\dfrac{x^2-1}{x^2-8x+7}=\lim\limits_{x\to1}\dfrac{(x-1)(x+1)}{(x-1)(x-7)}$$

$$=\lim\limits_{x\to1}\dfrac{x+1}{x-7}=\dfrac{1+1}{1-7}=-\dfrac{1}{3}$$

2)

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

$$=\lim\limits_{x\to1}\dfrac{x+1}{x-7}=\dfrac{1+1}{1-7}=-\dfrac{1}{3}$$

3)

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

4)

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