limits

**Laven** · Jul2-09, 04:24 AM

I'm whacked outta solving these problems just hanging around for more than 2 hours for each question but can't solve it.I think it's out of my capability.LOL.
I hope you will solve these problems.I need your help.The questions are...
1) lim sq.rt x[sqrt x - sqrt (x-a)]
x-->infinity

2) lim [tan x - sin x]/x^3
x--> 0

Thanks in advance.

dx · Jul2-09, 06:24 AM

You must show your attempt to receive help.

**HallsofIvy** · Jul2-09, 06:44 AM

You say you've been working on it for two hours (which is not really that much) so show what you have done in those two hours.

**Laven** · Jul3-09, 02:49 AM

Yea you both are right so that i can catch where my mistake is but it's so tough to write mathematical notations overhere.Could you plz give me a hint how can i write the mathematical notations?

**Office_Shredder** · Jul3-09, 03:00 AM

http://www.physicsforums.com/showthread.php?t=8997

It's a little bit of work to learn how to do latex, but then you can ask all your questions here much easier

**Laven** · Jul3-09, 06:10 AM

thanks office_shredder i got the idea to use latex.

Actually,for 1st question, i tried is a bit different one but similar to tht ones.The above questions seems complicated rather than i did.I gave yo cuz i get more idea to solve the question i tried.

Ok the question is [similar to tht one]

$LaTeX Code: \\lim_{x\\rightarrow \\infty} \\sqrt{3x}-\\sqrt{x-5}$

= $LaTeX Code: \\lim_{x\\rightarrow \\infty} (\\sqrt{3x}-\\sqrt{x-5}) X (\\sqrt{3x}+\\sqrt{x-5})$

**HallsofIvy** · Jul3-09, 06:42 AM

Originally Posted by Laven

thanks office_shredder i got the idea to use latex.

Actually,for 1st question, i tried is a bit different one but similar to tht ones.The above questions seems complicated rather than i did.I gave yo cuz i get more idea to solve the question i tried.

Ok the question is [similar to tht one]

$LaTeX Code: \\lim_{x\\rightarrow \\infty} \\sqrt{3x}-\\sqrt{x-5}$

= $LaTeX Code: \\lim_{x\\rightarrow \\infty} (\\sqrt{3x}-\\sqrt{x-5}) \\times (\\sqrt{3x}+\\sqrt{x-5})$

No, you cannot just multiply by something without changing the value. You can multiply and divide by the same thing:
$LaTeX Code: \\left(\\sqrt{3x}- \\sqrt{x-5}\\right)\\frac{\\sqrt{3x}+ \\sqrt{x- 5}}{\\sqrt{3x}+ \\sqrt{x- 5}}$
$LaTeX Code: = \\frac{3x- (x- 5)}{\\sqrt{3x}+ \\sqrt{x- 5}}= \\frac{2x+ 5}{\\sqrt{3x}+ \\sqrt{x- 5}}$

Now, to take the limit as x goes to infinity, divide both numerator and denominator by x, remembering that x will become x² inside the square roots.

**Laven** · Jul6-09, 05:42 AM

Yea i know tht hallsofivy actually tht was my true mistake because i was just trying using latex & i can't get it well.

actually.the process i'm gonna do is tht for the question
$LaTeX Code: <BR>(\\sqrt{3x}-\\sqrt{x-5})<BR>$

is....
I multiplied it by conjugate on both denominator and numerator sides & i got
$LaTeX Code: \\frac{2x+5}{\\sqrt{3x}+\\sqrt{x-5}}$
then i did 1st derivative on both numerator and denominator i got,

$LaTeX Code: \\frac{4\\sqrt{x(x-5)}}{\\sqrt{3(x-5)}+\\sqrt{x}}$

Again on 2nd derivative I got,
$LaTeX Code: \\frac{4x-10}{\\sqrt{3x}+\\sqrt{x-5}}$

So,i can't converge this answer.Anyone have idea to solve this question.Where am i wrong?could yo plz point it out?

**Laven** · Jul6-09, 05:55 AM

For the 2nd question i.e $LaTeX Code: \\lim_{x\\rightarrow0}(tan{x}-sin{x})$
1stly i find its derivative [on both sides numerator and denominator] i got,
$LaTeX Code: \\frac{1-cos^3{x}}{3x^2cos^2{x}}$
After further simplification with this i can't reach to the final answer.

Again on next way i change the trigonometric terms into half angle rule using formulae either way i can't get it.
Is there next way to solve this problem?

**slider142** · Jul7-09, 08:05 PM

Originally Posted by Laven

For the 2nd question i.e $LaTeX Code: \\lim_{x\\rightarrow0}(tan{x}-sin{x})$
1stly i find its derivative [on both sides numerator and denominator] i got,
$LaTeX Code: \\frac{1-cos^3{x}}{3x^2cos^2{x}}$
After further simplification with this i can't reach to the final answer.

Again on next way i change the trigonometric terms into half angle rule using formulae either way i can't get it.
Is there next way to solve this problem?

Since the denominator is a 3rd degree monomial, it is 0 as x approaches 0 up to the 3rd derivative, which should be a constant. Thus, you should look for the limit at the third application of L'Hopital's rule. Take care that the numerator remains 0 at each step as well, otherwise it would be invalid to apply L'Hopital's rule.