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. 👁 69
asked by Matt

Respond to this Question

First Name

Your Response

Similar Questions

  1. Health

    I posted this question yesterday and I got confused. The determinant of total energy requirement that may most appropriately be manipulated in a weight loss program is: 1. basal metabolic rate 2. physical activity 3. dietary

    asked by Lana on April 20, 2007
  2. For all tutors

    Is this just an American website. I am from Toronto Canada - posted a question yesterday and have not received a response. Just checking. Thanks.

    asked by Karen on October 18, 2007
  3. math

    i posted this question yesterday. i really want to know how to do it. John bought n objects, each costing n cents. The change he received from his $10 bill was less than $1 and consisted of only two denominations of coins. What

    asked by Diana on May 19, 2011
  4. Integral calculus

    Yesterday I posted this number that I don't understand. There was a mistake in the question,this is the right one: ∫ (5^(3/x) - (x^8 + 9)^1/3 ) / 3x^2 will u, for sub. be = x^8 + 9? Then du is = 8x^7 dx x^7 dx = 1/8 du Thank

    asked by Tommy on April 20, 2012
  5. Science

    This question I posted yesterday as well, but received no help. There wasn't any info about it in my book, so I'm wonderig if you could help me with it. Why might scientists want to figure out a way to promote cell division in

    asked by Sara on May 4, 2010
  6. Algebra II

    Simplify: x^5*x^pi Answer: x^8.14 Is this correct? I know that this is a repost of a question I posted yesterday, and I appreciate Damon looking at it, but I don't think I was clear in my typing of the question. Thanks.

    asked by Lucy on March 30, 2008
  7. To: Writeacher or any other english helper

    I had posted a question earlier about my essay but have received no help? is there no one who can help me? If you haven't seen the movie or read the book then i understand but please help if you can, really appreciated:)

    asked by Trisha on November 9, 2016
  8. math

    Posted by mota THE MASS OF THE SUN IS ABOUT 10^27 METRIC TONS , OR 10^30 KILOGRAMS . HOW MANY KILOGRAMS ARE IN ONE METRIC TON ? ----------------------- math - drwls 1000. ----------------------- explain please i want to understand

    asked by mota on March 2, 2008
  9. For MathMate

    Hi! So i posted a question yesterday and I worked on but before i submit my answer do you think you can check my answers? 1. 7x+3y= pi 4x-6y= pi^2 2. 2x+3y= 0 4x+6y= 0 2 3. x+3y=1 4x+ 6y= 1 4. x+y=5 x+2y=10 5. 2x-3y=5 4x-6y=10 1.U

    asked by Completely lost?! on September 11, 2009
  10. Math(Please help)

    I know I posted this yesterday but I really do not understand this and I did not quite understand the comment on what to do. Find the real solutions of the equation. x = 2 sqrt6x-36

    asked by Hannah on September 20, 2010
  11. Math

    I know I posted this yesterday but I really do not understand this and I did not quite understand the comment on what to do. Find the real solutions of the equation. x = 2 sqrt6x-36

    asked by Hannah on September 20, 2010

More Similar Questions