Solving Limits: Step-by-Step Instructions

[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]

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

 

[Townsend]

You have a lot of questions...thats ok but you need to read

https://www.physicsforums.com/showthread.php?t=44101

If you want to get help...at the very least write down what you think the solution to each one is...

 

[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!