Find the value of this limit

1. Dec 24, 2013

shivam01anand

1. The problem statement, all variables and given/known data

Find the value of the limit of the following two questions when x is tending to zero.

2. Relevant equations

[ (tan x)/(x) ]
[ (sin x ) / ( x ) ]

Where [ ] is the gint function..

3. The attempt at a solution

If the question was w/o gint then both these limits were 1 (right?)

But for gint have to check the graph of

y=x , y=sinx ,y=tanx,

so graph of tanx is above x which is above sinx ( x just after 0 )

Why is this wrong?

2. Dec 24, 2013

haruspex

3. Dec 24, 2013

shivam01anand

unfortunately i dont have the answers with me atm but ill post as soon as i get my hands on them.

here's another question similar to this{pointless to make a new post out of it}

[x^2/sinxtanx] where again [] is the gint function.

so i look at it like two functions.

(x/sinx)(x/tanx) one is tending to just below 0 while the other just greater than 1 but what will happen with the gint :x

Help!

4. Dec 24, 2013

haruspex

You mean, one is tending to just below 1 while the other just greater than 1, right?
Yes, separating them like that is not going to help.
It might help to boil the trig functions down to one trig function. I tried replacing the tan with sin/cos, but that didn't help. Do you know the "tan-half-angle" formulae, i.e. how sin(x), cos(x), tan(x) can be written in terms of t = tan(x/2)?

5. Dec 25, 2013

shivam01anand

You mean sin2x= 2tanx/1+tan^2(x)?

Dont really think this is helping :X

6. Dec 25, 2013

haruspex

It works for me. Please post what you get. (I wrote u = x/2, t = tan(u), and converted all to an expression using t and u.)

7. Dec 28, 2013

shivam01anand

okay you're right it worked for this question and certain similar question related to gint and sinx in it

Thanks!

Will keep this in mind :D

8. Dec 29, 2013

vanhees71

These are typical examples for the use of de L'Hospital's rule or (equivalently) doing a series expansion of numerator and denominator around the limit of $x$ (i.e., here around $x=0$).

9. Dec 30, 2013

shivam01anand

how does one apply L'h rule when the functions like gint/ fraction etc are involved?

10. Dec 30, 2013

Staff: Mentor

If the functions are not differentiable, that is not possible in a direct way. It can still help in some way (that depends on the function), but you still have to consider the gint brackets.

11. Dec 31, 2013

vanhees71

12. Dec 31, 2013

Dick

I assume it's ceiling. You can also attack problems like this by looking at the taylor series of the differentiable part around x=0.

Last edited: Dec 31, 2013
13. Dec 31, 2013

shivam01anand

gint(x)= greatest integer equal to or less than x.
I'm sorry for not naming it aptly ( this is what the folks call it down here)

14. Dec 31, 2013

shivam01anand

Hmm "dick" i hear what you're saying.

Thank you.