Calculus, please check my answers!

1. Evaluate:
lim x->infinity(x^4-7x+9)/(4+5x+x^3)

0
1/4
1
4
***The limit does not exist.

2. Evaluate:
lim x->infinity (2^x+x^3)/(x^2+3^x)

0
1
3/2
***2/3
The limit does not exist.

3. lim x->0 (x^3-7x+9)/(4^x+x^3)
0
1/4
1
***9
The limit does not exist.

4.For the function g(f)=4f^4-4^f, which of the following statements are true?
I. lim f->0 g(f)=-1
II. lim f->infinity g(f)=-infinity
III. g(f) has 2 roots.

I only
***II only
III only
I and II only
I, II, and III

5. lim cot(3x)
x->pi/3

sqrt3
1
(sqrt3)/3
0
***The limit does not exist.

6. lim (cos(x)-1)/(x)
x->0
1
***0
(sqrt2)/(2)
-1
The limit does not exist.

7. lim cos(x)-x
x->0
***1
0
(sqrt3)/(2)
1/2
The limit does not exist.

8. Which of the following functions grows the fastest?
***b(t)=t^4-3t+9
f(t)=2^t-t^3
h(t)=5^t+t^5
c(t)=sqrt(t^2-5t)
d(t)=(1.1)^t

9. Which of the following functions grows the fastest?
f(t)=2^t-t^3
a(t)=t^5/2
e(t)=e
g(t)=3t^2-t
***b(t)=t^4-3t+9

10. Which of the following functions grows the fastest?
***g(t)=3t^2-t
i(t)=1m(t^100)
e(t)=e
c(t)=sqrt(t^2-5t)
a(t)=t^5/2

11. Which of the following functions grows the slowest?
b(t)=t^4-3t+9
f(t)=2^t-t^3
h(t)=5^t+t^5
***c(t)=sqrt(t^2-5t)
d(t)=(1.1)^t

12. Which of the following functions grows the least?
g(t)=3t^2-t
i(t)=1n(t^100)
e(t)=e
c(t)=sqrt(t^2-5t)
***a(t)=t^5/2

13. Which of the following functions grows the slowest?
j(t)=1/4 1n(t^200)
a(t)=t^5/2
***i(t)=1n(t^100)
g(t)=3t^2-t
b(t)=t^4-3t+9

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  1. 1. Yep, the limit approaches infinity

    2. Nope, the limit is zero.
    lim (2^x+x^3)/(x^2+3^x) as x-> infinity
    We'll use L'Hopital's rule here (I hope it was already taught in your class). It is used when you get the form 0/0 or infinity/infinity when the x value is substituted. Just get the separate derivatives of numerator and denominator (actually we'll use the rule 4 times to get the ff):
    lim (2^x / 3^x)*[(ln(2)/ln(3))^4] as x->infinity
    let C = [(ln(2)/ln(3))^4] and we'll take it outside the limit, and we can rewrite the exponents as
    C*lim (2/3)^x
    Note that 2/3 = 0.66 which is greater than zero but less than 1, and when that number is raised to infinity, you'll approach zero.

    3. Yep, the limit is 9.

    4. I and II only are true.

    5. Yep, limit does no exist (it approaches both (+) and (-) infinity from both sides)

    6. Yep, it's zero.

    7. Yep, it's 1.

    For #s 8-13, I'm sorry I cannot help you, I can't remember how to determine which equation grows the fastest without using graphs. ^^;

    Anyway, I hope this helps~ :3

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  2. Thank you so much for not only verifying my answers, but also for your great explanations! I really appreciate it! :)

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