What is the value of these two limits when x tends to zero?

[shivam01anand]

Homework Statement

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

Homework Equations

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

Where [ ] is the gint function..

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 )


So 1st q answer=1

2nd q answer=0

Why is this wrong?

 

[haruspex]

I agree with your answers. What are the given answers?

 

[shivam01anand]

unfortunately i don't 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!

 

[haruspex]

unfortunately i don't 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!

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)?

 

[shivam01anand]

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


Dont really think this is helping :X

 

[haruspex]

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


Dont really think this is helping :X

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.)

 

[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

 

[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 $x=0$).

 

[shivam01anand]

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

 

[mfb]

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.

 

[vanhees71]

Ok, I've overlooked this "gint function". What the heck is it? I've never heard about it. Is this something like the floor or ceiling functions?

http://en.wikipedia.org/wiki/Floor_function

 

[Dick]

Ok, I've overlooked this "gint function". What the heck is it? I've never heard about it. Is this something like the floor or ceiling functions?

http://en.wikipedia.org/wiki/Floor_function

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.

 

[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)

 

[shivam01anand]

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

Thank you.