当x趋近于pai lim[sin(3x)/tan(5x)]

当x趋近于pai lim[sin(3x)/tan(5x)]
当x趋近于pai lim[sin(3x)/tan(5x)]

当x趋近于pai lim[sin(3x)/tan(5x)]
lim【x→π】[sin(3x)]/[tan(5x)]
=lim【x→π】[-sin(3x-3π)]/[tan(5x-5π)]
=lim【x→π】-(3x-3π)/(5x-5π) 【等价无穷小代换】
=lim【x→π】-3/5 【上式洛必达法则得到的】
=-3/5

sin(3x)/tan(5x)=sin(3x)/[sin(5x)/cos(5x)]=sin(3x)cos(5x)/sin(5x)
因为x趋近于π，故3x趋近于3π，5x趋近于5π，3π和5π相差2π，刚好是sinx的周期，故此时
sin(3x)=sin(5x)
lim[sin(3x)/tan(5x)]=lim[sin(3x)cos(5x)/sin(5x)]=limcos(5x)(x趋近于π)=cos5π=cosπ=-1