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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
=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
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