Vtyjkyuk

I am a kid trying to teach myself Calculus in order to prepare for next year.

I have the expression

$$lim_xto 2fraccos(frac pi x)x-2 $$

There is a hint that says to substitute t for $(frac pi2 - frac pi x)$ and WolframAlpha evaluates this expression as $frac pi4$. However, I got the answer of $1$. Can someone clarify the steps to solving this problem.

HINT:

$$cosdfracpi x=sinleft(dfracpi2-dfracpi xright)=sindfracpi(x-2)2x$$

Now set $dfracpi(x-2)2x=y$ and use $lim_hto adfracsin(h-a)h-a=1$

Here is my solution using only $lim_x to 0 fracsin(x)x = 1$, using the methods the questions asks for (the $t$ substitution)
$$lim_x to 2 fraccos(pi/x)x-2$$
$$ = lim_x to 2 fracsin(fracpi2 - fracpix)x-2$$
substitute $t= fracpi2 - fracpix$, $;;x = frac2 pipi-2 t$
$$ = lim_t to 0 fracsin(t)frac2 pipi-2 t-2$$
$$ = lim_t to 0 fracsin(t)frac2 pi - 2(pi-2 t)pi-2 t$$
$$ = lim_t to 0 frac(pi-2 t)sin(t)2 pi - 2(pi-2 t)$$
Substitute $r = pi-2 t$, $;;t = fracpi-r2$
$$ = lim_r to pi fracrsin(fracpi-r2)2(pi - r)$$
$$ = frac14lim_r to pi fracrsin(fracpi-r2)fracpi-r2$$
Substitute $k = fracpi-r2$, $;;r= pi-2k$
$$ = frac14lim_k to 0 frac(pi-2k)sin(k)k$$
$$ = frac14bigg[lim_k to 0 (pi-2k)*lim_k to 0fracsin(k)kbigg]$$
$$=fracpi4$$
I unfortunately only got this after an answer was accepted, as messing around with a limit this much takes a while and involved some trial and error before I could figure out what to do... there might be a way to condense this proof, but I always jump right into these things by making substitution after substitution until I get to a form that I know.

$$lim_xto2fraccos(π/x)x-2$$
Which leads us to an Indeterminate form i.e.,(0/0) form ,
so apply L-Hospital rule ,

$$lim_xto2[-sin(π/x)-π/x^2]
=lim_xto2 fracπx^2
=fracpi4$$

Let $x=2+y$ with $yne 0$.We have $x-2=y$ and $ pi/x=pi/(2+y)=pi/2-pi y/(4+2 y) . $ So $cos pi/x=sin pi y/(4+2 y)=(1-f(y)(pi y/(4+2 y)$ where $lim_yto 0 f(y)=0 . $ Hence $(cos pi/x)/(2-x)= (1-f(y)(pi/(4+2 y)).$

Let $dfrac1x=y$ then
beginalign
limlimits_x to 2 fraccosleft(fracpixright)x-2
&= limlimits_y to frac12 fracsinleft(fracpi2-pi yright)dfrac1y-2\
&= limlimits_y to frac12 fracsinpileft(frac12- yright)pileft(frac12- yright)dfracleft(frac12- yright)1-2ypi y\
&= colorbluedfracpi4
endalign