# Cal II Question

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

Townsend
Tonya Miller said:
Explain how to complete these problems:
lim (x-1)^1/2 - 2/ x^2-25
x 5
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....

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

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.

lim x^3/2 + 5x - 4/ x ln x
x infinity
I need to know what you mean here too.

lim tan x ln sin x
x pie/2-

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.

lim (cos x)^x+1
x 0
Why not just plug in 0 directly? Looks like 2 to me.

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

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.

steven187
hello there

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

steven

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

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

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

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

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

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

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

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

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

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

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

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