lim[xsin(PI/x)+(PI/x)sinx]=?x->无穷大

1个回答

  • 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