lim[xsin(PI/x)+(PI/x)sinx]=?x->无穷大

lim_{x->无穷大}[xsin(PI/x)]
= lim_{x->无穷大}[PIsin(PI/x)/[PI/x]]
= PIlim_{x->无穷大}[sin(PI/x)/[PI/x]]
= PI
lim_{x->无穷大}[(PI/x)sin(x)]
= PIlim_{x->无穷大}[sin(x)/x]
因,|sin(x)| 无穷大}[1/x] = 0.
所以,
lim_{x->无穷大}[(PI/x)sin(x)]
= PIlim_{x->无穷大}[sin(x)/x]
= PI*0
= 0.
因此,
lim_{x->无穷}[xsin(PI/x) + (PI/x)sinx]
= lim_{x->无穷大}[xsin(PI/x)] + lim_{x->无穷大}[(PI/x)sin(x)]
= PI + 0
= PI