1ãæ ¹æ®
ææ ¼ææ¥ä¸å¼å®çarctana-arctanb=1/(1+ξ²)·(a-b)
å
¶ä¸ï¼Î¾å¨aä¸bä¹é´ï¼
â´arctan(Ï/n)-arctan[Ï/(n+1)]
=1/(1+ξ²)·[Ï/n-Ï/(n+1)]
=Ï/[n(n+1)(1+ξ²)]
å
¶ä¸ï¼Î¾å¨Ï/(n+1)ä¸Ï/nä¹é´ï¼
â´åå¼=limn²Â·Ï/[n(n+1)(1+ξ²)]
=limÏ/[(1+1/n)·(1+ξ²)]
=Ï
ãâµlim(1+ξ²)=1ã
2ãæ ¹æ®ææ ¼ææ¥ä¸å¼å®ç
e^a-e^b=e^ξ·(a-b)
å
¶ä¸ï¼Î¾å¨aä¸bä¹é´ï¼
â´e^[1/(2x-1)]-e^[1/(2x+1)]
=e^ξ·[1/(2x-1)-1/(2x+1)]
=2e^ξ/(4x²-1)
å
¶ä¸ï¼Î¾å¨1/(2x-1)ä¸1/(2x+1)ä¹é´ï¼
â´åå¼=limx²Â·2e^ξ/(4x²-1)
=lim2e^ξ/(4-1/x²)
=1/2
ãâµlime^ξ=e^0=1ã