# Solving Limits: Step-by-Step Instructions

• Tonya Miller
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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

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.

hello there

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

steven

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

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

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

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

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

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

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

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

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

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

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!

## 1. What are limits in mathematics?

Limits in mathematics refer to the value that a function approaches as its input approaches a certain value. It is used to determine the behavior of a function near a particular point.

## 2. How do you solve limits?

To solve a limit, you need to follow the rules of limit evaluation, such as direct substitution, factoring, and rationalizing the denominator. You may also need to use special techniques, such as L'Hopital's rule or trigonometric identities, depending on the type of limit.

## 3. What are the steps to solving limits?

The steps to solving limits are as follows:

1. Identify the type of limit (e.g. indeterminate form, one-sided limit, etc.)
2. Try to evaluate the limit using direct substitution
3. If direct substitution does not work, try to simplify the function using algebraic manipulation
4. If the limit is still indeterminate, use special techniques such as L'Hopital's rule or trigonometric identities
5. Check for any restrictions on the domain that may affect the limit
6. Write the final answer with the appropriate notation

## 4. What are some common mistakes when solving limits?

Some common mistakes when solving limits include:

• Forgetting to check for restrictions on the domain
• Incorrectly applying limit rules or special techniques
• Not simplifying the function enough before trying to evaluate the limit
• Forgetting to write the final answer with the appropriate notation (e.g. infinity or a specific value)

## 5. Can limits be solved using a calculator?

Yes, some calculators have the ability to solve limits. However, it is important to understand the steps and concepts behind solving limits manually before relying on a calculator. Also, not all limits can be solved using a calculator, so it is important to know when and how to use one effectively.

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