Calculate the lim x→0 f(x)#calculas

[wind gust]

#1020426321261756536

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[crimson ruin]

Hmm well my idea Is to use Feynman’s technique

Never mind not a good idea

I’m overthinking it this is just (u’v-uv’)/v^2

Here u=sin(y) and v=y

You can use integration by parts to verify it

[crimson ruin]

(u/v)’

Knowing that you can evaluate the integral from x to pi/2 and then take the limit to 0

[crimson ruin]

For the limit to 0 think of l’hôpital !

[hexed torrent]

L'Hopital's rule only works if both the functions in the numerator and denominator return 0 upon directly substituting the limit ryt?

[crimson ruin]

In my country we don’t study l’hôpital in college or in high school ( at least for now ) so I don’t know

But another way of looking at it is the rate of change

For example lim x->0 (ln(x+1))/x=1=1/1+0

Because ln(1+x)’=1/(1+x) so evaluating it in 0 is 1

In this case we can use this argument as well

[crimson ruin]

And in this case they both do

If I’m not mistaken

[hexed torrent]

hmmm

u get Jcosy/y - Jsiny/y^2

then integrate cosy/y

but how

[crimson ruin]

Not cos but sin

[hexed torrent]

oh wait

integral 1/y is Ln(y)

i'm stupid

[crimson ruin]

Its (u/v)’ with u=siny and v=y

[hexed torrent]

hmm

let me try once

[crimson ruin]

Integrate you get sin(y)/y

[hexed torrent]

huh

I don't know all the integration formulas unfortunately

[crimson ruin]

Well do you know the formula (u/v)’=(u’v-uv’)/v^2

?

[hexed torrent]

that's differentiation

Ik that

[crimson ruin]

Yes test it on sin(y)/y

With respect to y

[hexed torrent]

ycosy - siny/y^2

damn

so sin(y)/y

[crimson ruin]

Yes

You can apply l’hôpital then

[hexed torrent]

yep

to get cosy

[crimson ruin]

Yes

[hexed torrent]

wait

x is in the definite integral

so, sin(pi/2)/pi/2 - lim(x->0)siny/y

2/pi - 1

[crimson ruin]

Yep

[hexed torrent]

wait that's correct

wow

yay me

[crimson ruin]

Yes that’s correct

Gg

[hexed torrent]

yay

[wind gust]

Can you please provide me the full solution?

If you don’t mind

[crimson ruin]

Sure

[knotty python]

Just note that (y cos(y) - sin(y))/y^2 = (sin(y)/y)'. From that you can evaluate the integral using the fundamental theorem of calculus, then calculate the limit.

[wind gust]

Thanks buddy

[wind gust]

Thanks sir

[molten rivet]

@wind gust has given 1 rep to @knotty python

[knotty python]

You're welcome!

[wind gust]

Hi all dear sirs

Can anyone send me the proper written solution on paper?

[crimson ruin]

Hope this is clear enough

You don’t need the last part I just wanted to test different proofs

[wind gust]

Thanks a lot sir

[wind gust]

I had tried it that way already. Also you cannot directly subst for x as 0