# Cal II Question

1. Jul 27, 2005

### Tonya Miller

Explain how to complete these problems:

lim (x-1)^1/2 - 2/ x^2-25
x 5

lim x + 1-e^x/ x^3
x 0+

lim x^3/2 + 5x - 4/ x ln x
x infinity

lim tan x ln sin x
x pie/2-

lim (cos x)^x+1
x 0

lim (1+1/x)^5x
x infinity

2. Jul 27, 2005

### Townsend

I assume you mean...

$$\lim_{x -> 5} \frac{\sqrt(x-1) -2}{x^2-25}$$

To do this problem just multiply the top and bottom by the conjugate of the numerator....use the difference of square to factor the x^2-25 and you will have a common factor to take out....

On this one I really can't say without knowing exactly what is in the numerator and denominator... try using parentheses so I know how things are grouped.

I need to know what you mean here too.

I haven't brushed up on indeterminate forms but I believe this is a case where you can apply L.H....but not directly...write as $$ln(sin(x))/cot(x)$$ and then you have a 0/0....L.H. gives you...check my work...-cos(x)*sin(x) And this limit is clearly 0.

Why not just plug in 0 directly? Looks like 2 to me.

Well if you wrote it like

$$((1+ \frac{1}{x})^x)^5$$

And you know what the inside limit is....so did euler....then just raise that to the fifth power.

3. Jul 27, 2005

### steven187

hello there

do you want to prove the limit exists? or do you want to find the limit?

steven

4. Jul 28, 2005

### Tonya Miller

Could you show me step by step how to handle these problems? I need to find the limit.

#2 lim (x+1-e^x)/(x^3)
x=>0+

#3 lim (x^3/2 +5x-4)/(xlnx)
x=>infinity

5. Jul 28, 2005

### Tonya Miller

Could you show how to do these integrals?

infinity
S x/x^4+9 dx
neg.infinity

0
S 1/(x-8)^2/3 dx
neg.infinity

0
S x/(4-x^2)1/2 dx
-2

pie
S 1/(1-cos x) dx
0

6. Jul 28, 2005

### Tonya Miller

Could show step by step how to do these power series converge?

infinity
E n/(n^2 + 1) x^n
n=2

E [(1)(3)(5)(7)...(2n-1)]/[(3)(6)(9)(12)...(3n)] (x-2)^n

7. Jul 28, 2005

### Tonya Miller

Could you show in details how to converge or diverge these problems?

1) {n/n+1}

2) E 2^n/(n^2)

3) E (n!)^2/ (2n)!

4) E (n 3^2n)/ (5^n-1)

5) E (-1)^n+1 n/(n^2 +4)

infinity
6) E (-1)^n n/(ln n)
n= 2

8. Jul 28, 2005

### Tonya Miller

Could you show me how to find dy/dx for the following problems?

1) x=t^3, y= t^2

2) x = sec t + 2, y = tan t -1

9. Jul 28, 2005

### Tonya Miller

Could you show me how to find the equation of the tangent line to this function?

1) x = (t)^1/2 , y = 3t + 4, t = 4

10. Jul 28, 2005

### Tonya Miller

Could you show step by step how to integrate these problems?

1) S 1/ (x)^1/2 [(x)^1/2 + 1] dx

2) S x atn x^2 dx

3) S x 4^-x2 dx

4) S x^2/ x+ 2 dx

5) S (e^2x + e^3x)^2/ (e^5x) dx

6) S csc(1+ cot(x)) dx

11. Jul 28, 2005

### Tonya Miller

Could show how to integrate these problems?

1) S (x^2+3x+5) e^3x dx

2) S e^4x sin(5x) dx

3) S sec^3 (3x) dx

4) S sin^2 4x cos^2 4x dx

5) sec x/ (cot^5 x) dx

6) (4-x^2)^1/2 / (x^2) dx

12. Jul 28, 2005

### Tonya Miller

Could you show how to integrate these problems?

1) S 3x-5 / (1 + x^2)^1/2 dx

2) S 5x^3- 3x^2 + 7x - 3/ (x^4 + 2x^2 + 1) dx

3) S 5x^2 + 11x +17/ (x^3 + 5x^2 + 4x + 20) dx

13. Jul 28, 2005

### Tonya Miller

Could you show step by step how to find the derivatives for these problems?

1) y = ln (x/ 3x + 5))^4

2) y = (ln (x)^1/2)^1/2

3) y = 5^3x + (3x)^5

4) y = x^5x+1

5) y = (sin x )^ cos x

14. Jul 28, 2005

### Tonya Miller

Could you show me step by step how to handle these problems? I need to find the limit.

#2 lim (x+1-e^x)/(x^3)
x=>0+

#3 lim (x^3/2 +5x-4)/(xlnx)
x=>infinity

Could you show how to do these integrals?

infinity
S x/x^4+9 dx
neg.infinity

0
S 1/(x-8)^2/3 dx
neg.infinity

0
S x/(4-x^2)1/2 dx
-2

pie
S 1/(1-cos x) dx
0

15. Jul 28, 2005

### Tonya Miller

Could show step by step how to do these power series converge?

infinity
E n/(n^2 + 1) x^n
n=2

E [(1)(3)(5)(7)...(2n-1)]/[(3)(6)(9)(12)...(3n)] (x-2)^n

Could you show in details how to converge or diverge these problems?

1) {n/n+1}

2) E 2^n/(n^2)

3) E (n!)^2/ (2n)!

4) E (n 3^2n)/ (5^n-1)

5) E (-1)^n+1 n/(n^2 +4)

infinity
6) E (-1)^n n/(ln n)
n= 2

16. Jul 28, 2005

### Townsend

17. Jul 29, 2005

### HallsofIvy

Tonya, you have listed a huge number of what look like homework problems (and, if so, should be in the homework section) without showing any of your own work. Surely you must have some ideas about these problems- showing what you have done will help us give you hints and tips. If you really have absolutely no idea how to even begin, you need more help than we can give- the best thing you can do is go to your teacher and let him/her know about your difficulty!