Calculus help, please!

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. no ideas on any of these?

    #1 as x gets huge, f(x) just looks like x^4/x^3

    #2 as x gets huge, exponentials outpace any polynomial, so f(x) just looks like 2^x/3^x = (2/3)^x

    #3 as x->0, you have 9/0

    #4 II only

    #5 cot(pi) = 1/tan(pi)

    #6 use L'Hospital's Rule to get -sin(x)/1

    #7 no trick at all

    #8 exponentials outpace polynomials
    larger base grows faster

    #9 ditto

    #10 ditto (I think - here there be typos?)

    #11 powers are slower than exponentials.
    lower powers are slower than higher powers

    #12 logs are even slower than roots

    #13 ditto

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    posted by Steve

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