Question

这道高数极限题怎么做？

Answer 1

用等价无穷小
1-cosx～1/2x^2
原式＝x^2/（1/2x^2）＝2

Answer 2

x→0时，sinx∽x
方法一:
0/0型，运用洛必达法则
lim(x→0) x²/(1–cosx)
=lim(x→0) 2x/sinx
=2
方法二:
lim(x→0) x²/(1–cosx)
=lim(x→0) x²/[2sin²(x/2)]
=lim(x→0) x²/[2·(x/2)²]
=2
方法三:
在x=0处，cosx带皮亚诺余项的泰勒展开为
cosx=1–x²/2＋x^4/4！＋o(x^4)
=1–x²/2＋o(x²)
lim(x→0) x²/(1–cosx)
=lim(x→0) x²/[1–(1–x²/2＋o(x²))]
=lim(x→0) 1/(1/2)
=2