limits (1 Viewer)

[senso]

In a few of the recent past hsc papers, there has been a question like
what is the limit as x approaches 0 of (sinx)/5x
in the success one books, they always say that sin5x/5x =1 (or also 3x/sin3x =1 for example)
But how does this all work? I can get answers for these basic ones, but don't know what you are actually meant to do for limits questions

 

[Slidey]

lim[(sinx)/5x]=lim[(1/5)*sinx/x]=(1/5)*lim[sinx/x]=(1/5)*1
It's a limit you're meant to know.

If you want more information do a google search on "L'Hopital" and "limit".

EDIT: Note L'Hospital's rule is outside syllabus scope. All you need to know is that it works.

Basically, if you have the limit of f(x)/g(x), where both functions return zero when the limit is taken, then the limit is equal to the limit of their derivatives, like so:
Where lim f(x) = lim g(x) = 0, lim f(x)/g(x)=lim f'(x)/g'(x).

example:
as x -> 0 lim sinx/x = lim cosx/1 = lim cosx = 1

 

[Slidey]

example: sin2x/x
sin2x=2sinxcosx
so, as x->0 lim2cosx*(sinx/x)=2limcosx * limsinx/x = 2 * 1 * 1

 

[mojako]

probably Slide Rule has made the point but Im too lazy to read it..

It can be shown by the wonder of mathematics that
sin(x)<= x <= tan(x), x in radians

It can also be shown that for small x,
i.e. as x->0,
sin(x) == x == tan(x) where == is approximate equal to

Hence,
lim sin(x)/x = 1
x->0

This has the implication that
lim sin(kx)/kx = 1 , where k is any constant
kx->0

For not-very-large k, this means that when x->0, kx->0 also.
Hence,
lim sin(kx)/kx = 1, for not-very-large k
x->0

Similarly,
lim tan(x)/x = 1
x->0

also,
as x->0, cos(x)->1
since cos(x)=sin(x)/tan(x) and sin(x)==tan(x)

 

[Slidey]

It can be shown by the wonder of mathematics that
sin(x)<= x <= tan(x), x in radians

let x=8pi radians

0 <= ~25 <= 0

Uh, no?

 

[mojako]

let x=8pi radians

0 <= ~25 <= 0

Uh, no?

oh well,
I should place restriction on the value of x...

But since I'm a very kind person I dont wanna restrict x to imprisonment..
just let it wander free in the universe...

 

[Slidey]

That's commendable of you Mojako.

 

[senso]

makes more sense now.. thanks guys