Answer
The limit does not exist
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{x\rightarrow 2\pm} \cos\left(\dfrac{\pi}{2}(x-[x])\right)$
Compute the value of $\cos\left(\dfrac{\pi}{2}(x-[x])\right)$ for values of $x$ approaching $2$ from left and right:
$\cos\left(\dfrac{\pi}{2}(1.9-[1.9])\right)\approx 0.15643447$
$\cos\left(\dfrac{\pi}{2}(1.99-[1.99])\right)\approx 0.0.015707317$
$\cos\left(\dfrac{\pi}{2}(1.999-[1.999])\right)\approx 0.0015707957$
$\cos\left(\dfrac{\pi}{2}(1.9999-[1.9999])\right)\approx 0.0001570796$
$\cos\left(\dfrac{\pi}{2}(2.1-[2.1])\right)\approx 0.98768834$
$\cos\left(\dfrac{\pi}{2}(2.01-[2.01])\right)\approx 0.99987663$
$\cos\left(\dfrac{\pi}{2}(2.001-[2.001])\right)\approx 0.99999877$
$\cos\left(\dfrac{\pi}{2}(2.0001-[2.0001])\right)\approx 0.99999999$
Therefore we get:
$\displaystyle\lim_{x\rightarrow 2^{-}} \cos\left(\dfrac{\pi}{2}(x-[x])\right)=0$
$\displaystyle\lim_{x\rightarrow 2^{+}} \cos\left(\dfrac{\pi}{2}(x-[x])\right)=1$
As the left hand limit and the right hand limit are not equal, the limit of the function in $x=2$ does not exist.
