Question

求下列二元函数极限

十分感恩!!!

Answer 1

lim(x→0) [2－∨(xy＋4)]/(xy)
=lim(x→0) －1/[2∨(xy＋4)]
=－1/4
lim(y→0，x→2) sin(xy)/y
=lim(y→0) sin2y /y
=lim(y→0) 2cos2y
=2
lim(x→∞，y→2) (1＋y/x)^[x²/(x＋y)]
=lim(x→∞) (1＋2/x)^[x²/(x＋2)]
=lim(x→∞) e^｛x²/(x＋2) ln(1＋2/x)｝
=lim(x→∞) e^｛ ln(1＋2/x) /[(x＋2)/x²]｝
=lim(x→∞) e^[ln(1＋2/x) /(1/x)]
=lim(x→∞) e^[1/(1＋2/x) (－2/x²) /(－1/x²)]
=lim(x→∞) e^[2/(1＋2/x)]
=e²
lim(x→0，y→0) [1－cos(x²＋y²)]/[(x²＋y²)e^(x²y²)]
=lim(x→0) (1－cosx²)/x²
=lim(x→0) 2xsinx²/2x
=0