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

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

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

1. The limit does not exist.

2. The limit is 2/3.
3. The limit is 9.
4. II only.
5. The limit does not exist.
6. The limit is 0.
7. The limit is 1.
8. b(t) = t^4-3t+9.
9. b(t) = t^4-3t+9.
10. g(t) = 3t^2-t.
11. c(t) = sqrt(t^2-5t).
12. a(t) = t^5/2.
13. i(t) = ln(t^100).

To evaluate limits, we need to analyze the behavior of the function as the input approaches a certain value, usually infinity or zero. We can do this by simplifying the function or applying limit rules.

For questions 1-3, we need to find the limit of the given function as x approaches a specified value.

1. To evaluate lim x->infinity(x^4-7x+9)/(4+5x+x^3):
We divide both the numerator and denominator by the highest power of x (x^3) to simplify the expression. This gives us (x^4/x^3 - 7x/x^3 + 9/x^3)/(4/x^3 + 5x/x^3 + x^3/x^3). As x approaches infinity, all terms with x in the denominator become 0. Therefore, the limit becomes (0 - 0 + 0)/(0 + 0 + 1) = 0/1 = 0. So the correct answer is 0.

2. To evaluate lim x->infinity (2^x+x^3)/(x^2+3^x):
As x approaches infinity, the exponents dominate the terms with x^3, x^2, and 3^x. Therefore, we can simplify the expression to (2^x/3^x)/(3^x/3^x). Dividing both the numerator and denominator by 3^x, we get (2/3)^(x-x). Simplifying further, we have (2/3)^0 = 1. So the correct answer is 1.

3. To evaluate lim x->0 (x^3-7x+9)/(4^x+x^3):
As x approaches 0, we can simplify the expression by removing the terms with x^3 and 4^x, as they tend to 0. We are left with (-7x+9)/1, which becomes 9 as x approaches 0. So the correct answer is 9.

For questions 4-13, we need to analyze the given functions.

4. For the function g(f) = 4f^4 - 4^f:
I. To find the limit as f approaches 0, we substitute 0 into the function and get g(0) = 4(0)^4 - 4^0 = 0 - 1 = -1. So the statement I is incorrect.
II. To find the limit as f approaches infinity, we observe that as f increases, the exponential term 4^f grows much faster than the polynomial term 4f^4. Therefore, the function approaches negative infinity. So the statement II is correct.
III. To find the number of roots, we need to solve the equation g(f) = 0. However, this equation cannot be easily solved analytically. So the statement III is uncertain. Therefore, the correct answer is II only.

5. To evaluate lim cot(3x) as x approaches pi/3:
We can rewrite cot(3x) as cosine(3x)/sin(3x), and then analyze the limit of each term separately. As x approaches pi/3, the cosine term approaches 1/2 and the sine term approaches sqrt(3)/2. However, dividing 1/2 by sqrt(3)/2 gives us 1/sqrt(3), which is not one of the given answer choices. Therefore, the limit does not exist.

6. To evaluate lim (cos(x)-1)/x as x approaches 0:
We can use a trigonometric identity to simplify the expression: (cos(x)-1)/x = -2(sin^2(x/2))/(2(x/2)). Canceling the common terms, we get -sin^2(x/2)/(x/2). As x approaches 0, sin(x/2) approaches 0, and the expression becomes 0/0. Therefore, we need to apply L'Hopital's rule by taking the derivative of the numerator and denominator separately.
Differentiating the numerator and the denominator, we get lim (cos(x)-1)/x = lim -sin(x)/(1) = 0/1 = 0. So the correct answer is 0.

7. To evaluate lim (cos(x)-x) as x approaches 0:
As x approaches 0, the cosine term approaches 1, and x approaches 0. Therefore, 1 - 0 = 1. So the correct answer is 1.

8-13: To determine which function grows the fastest or slowest, we need to compare their growth rates as the input variable increases.

8. The function b(t) = t^4 - 3t + 9 has the highest degree term as t^4, which grows faster than the other functions. Therefore, b(t) grows the fastest.

9. Similar to the previous question, the function b(t) = t^4 - 3t + 9 still has the highest degree term as t^4, so it grows the fastest.

10. The function g(t) = 3t^2 - t has the highest degree term as t^2, which grows faster than the other functions. Therefore, g(t) grows the fastest.

11. The function c(t) = sqrt(t^2 - 5t) has a square root term that grows slower than the other functions. Therefore, c(t) grows the slowest.

12. The function a(t) = t^5/2 has a fractional exponent with a lower degree than the other functions. Therefore, a(t) grows the least.

13. The function i(t) = 1n(t^100) has a logarithmic growth, which is slower than the other functions. Therefore, i(t) grows the slowest.