Question

lim[xsin(PI/x)+(PI/x)sinx]=?x->无穷大

Answer 1

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