12+ersinxlim例3.计算X4X-→0x1+ex解：当X→0时11C+er2+er2e-sinxsinxsinxtelim[=lim[=lim|十444X-0X->0X-→0+x中x1+ex1+exX（+）esinx=lim三x→0+x2+ex2+ersinxsinxlim=lim=2-1=1当x→0时44/xX->0X-0x1+er1+er所以，原极限为1.例3. 计算 1 4 0 2 sin lim[ ] | | 1 x x x e x x e → + + + 解： 1 4 0 2 sin lim[ ] | | 1 x x x e x x e → + + + + 当 时 + x → 0 1 4 0 2 sin = lim[ ] 1 x x x e x x e → + + + + 4 3 4 0 2 sin = lim[ ] 1 x x x x e e x x e + − − → − + + + 0 sin = lim x x x → + =1 当 x 0 时 → − 1 4 0 2 sin lim[ ] | | 1 x x x e x x e → − + + + 1 4 0 2 sin = lim[ ] 1 x x x e x x e → − + − + = − = 2 1 1 所以，原极限为1