Lim tan5x/sin3x X→∏ Lim( sinx-x*cosx)/(x2*sinx) X→0 Lim Lntan7x/ Lntan2x X→0+ Lim tan5x/sin3xX→∏Lim( sinx-x*cosx)/(x2*sinx)X→0Lim Lntan7x/ Lntan2x X→0+
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Lim tan5x/sin3x X→∏ Lim( sinx-x*cosx)/(x2*sinx) X→0 Lim Lntan7x/ Lntan2x X→0+ Lim tan5x/sin3xX→∏Lim( sinx-x*cosx)/(x2*sinx)X→0Lim Lntan7x/ Lntan2x X→0+
Lim tan5x/sin3x X→∏ Lim( sinx-x*cosx)/(x2*sinx) X→0 Lim Lntan7x/ Lntan2x X→0+
Lim tan5x/sin3x
X→∏
Lim( sinx-x*cosx)/(x2*sinx)
X→0
Lim Lntan7x/ Lntan2x
X→0+
Lim tan5x/sin3x X→∏ Lim( sinx-x*cosx)/(x2*sinx) X→0 Lim Lntan7x/ Lntan2x X→0+ Lim tan5x/sin3xX→∏Lim( sinx-x*cosx)/(x2*sinx)X→0Lim Lntan7x/ Lntan2x X→0+
不断运用洛比达法则.
(1) tan5x 5/cos²5x 5/cos²(5π) 5
lim ------- = lim ---------- = ------------ = --- = 5/3
x→π tan3x x→π 3/cos²3x 3/cos²(3π) 3
(2)
sinx-xcosx cosx-cosx+xsinx
lim ------------ = lim -----------------
x→0 x²sinx x→0 x²cosx+2xsinx
sinx
= lim ----------------
x→0 xcosx+2sinx
cosx
= lim -----------------
x→0 cosx-xsinx+2cosx
1
= --------
3-0
= 1/3
(3) ln(tan7x) 1/tan7x·1/cos²(7x)·7
lim ------------ = lim -------------------------
x→0+ ln(tan5x) x→0+ 1/tan5x·1/cos²(5x)·5
14/sin(14x)
= lim ---------------
x→0+ 10/sin(10x)
14/(14x)
= lim -----------
x→0+ 10/(10x)
= 1