# Calculus-Limits

Okay, i posted this question yesterday, however, I did not really understand the answer I received. If your the one who answered my question, could you please elaborate.
If not, could you try to answer this tough, for me, question. Thanks a lot.

lim
x-->0

Square root (1 + tan x) - Square root (1 + sin x) all over x^3.

Limits involving analytical functions are most easily calculated using series expansion techniques. In this case we want to calulate a limit for x--> 0, so the natural thing to do is to examine how the functions that are involved in this limit behave in the vicinity of x = 0.

In this problem you need to use that:

sin(x) = x - x^3/6 + term of order x^5

cos(x) = 1 - x^2/2 + term of order x^4

You find tan(x) by dividing sin(x) by cos(x).

Both tan(x) and sin(x) tend to zero for x -->0 so you need to know how the square root function behaves near 1. For y close to zero we have:

sqrt[1 + y] = 1 + 1/2 y + (1/2)(-1/2)/2 y^2 + (1/2)(-1/2)(-3/2)/6 y^3+ term of order y^4

You just insert the series expansion for the tan and sin in y and subrtact the two terms...

It is possible to calculate limits without using series expansions. But that's more tedious because ultimately you must make use of the way the fiunctions behave near zero. If you don't make use of the series expansions then the information contained in them must be derived in some form during that derivation.

Let me give some other examples:

Lim x-->0 sin(x)/x = 1

If you use that sin(x) = x + term of order x^3, then you immediately see that:

sin(x)/x = 1 + term of order x^2

Take limit x-->0 of both sides and you find the desired result.

Lim x-->0 (sin(x)-x)/x^3 = -1/6

Use that:

sin(x) - x = - x^3/6 + term of order x^5

Divide both sides by x^3:

(sin(x) - x)/x^3 =
- 1/6 + term of order x^2

Take the limit x-->0 to find te desired result.

You can also calculate the latter limit by applying L'Hopital's rule three times! :)

1. 👍 0
2. 👎 0
3. 👁 212

## Similar Questions

1. ### Math Riddle

What day would yesterday be if Wednesday was five days before the day after tomorrow? pLEASE HELP ME WITH THE RIDDLE i POSTED!!!!!! EARLIER waht materials could upumuse Alright already! I really wondered why you just didn't pick

asked by Angel67 on August 30, 2006
2. ### Social Studies

I got the question correct but I don't understand why the answer I chose was correct. (I eliminated the ones that I knew were wrong and guessed out of the two answers I had left) Here's the question: Which event contributed most

asked by Evalyn Carver on April 9, 2018
3. ### bobpursley, drwls. ms. sue Writeacher

When a second post is made to a question, the most recent is now at the bottom. Do any of you prefer the way it was, or do you like this better? I prefer to have the answers shown in the order they were posted. The second answer

asked by Lance on June 16, 2005
4. ### BIO

I really need help with this question, Damon gave me an answer, but I do not know if it is for this question or for the one posted after. This is the only question I need help on, I promise not to post any other questions after

asked by Andrea on July 7, 2010
5. ### chemistry

Determine the number of moles of a) N atoms in a sample of C7H5(NO2)3 that has the same number of O atoms as 12.6g C6H12O6 Someone posted a way to answer this question but i still don't understand

asked by tomi on September 20, 2008
1. ### math

Hi, I posted a question yesterday. How do I get to see if someone answered? Thanks, John

asked by John on June 3, 2016
2. ### Chemistry

How do i determine how many electrons are in the highest occupied energy level? For Example how many electrons are in the highest occupied energy level group 3A I answered this for you when you first posted yesterday (last night).

asked by Bryan on November 1, 2006
3. ### Math

Good evening. I posted a math question earlier that I never received any help on because someone posted something inappropriate. My new math problem is as follows: The area of a rectangular piece of a land is 3/16 square miles. If

asked by Donald on February 21, 2017
4. ### Chemistry

Hi, I posted yesterday with this lab question. "Calculate the molar amounts of NaOH used in the reaction with the HCl solution and with the HC2H3O2. I think I got the answer to that with the help that I received. I had 5 mL of

asked by Rachel on April 22, 2007
5. ### Math

I posted this problem the other day and someone answered it but when I checked my book I realized that I wrote the problem wrong. This morning when I posted I had put the wrong answer. The answer is suppose to be 1/8 but I don't

asked by Josh on January 23, 2010
6. ### Math

840 stickers were given to 42 children. ⅔ of the children were boys, and each of them received the same number of stickers. Each girl received twice as many stickers as each boy. How many stickers did each girl receive? Explain

asked by Anonymous on February 19, 2015