Question

高数求极限问题

请给出步骤

Answer 1

n--->∞时
原式--->n^2[π/n-π/(n+1)]
--->nπ/(n+1)
--->π。

Answer 2

lim(n->无穷 )n^2. (arctan(π/n)- arctan[π/(n+1)] )

consider
lim(x->无穷 ) x^2. (arctan(π/x)- arctan[π/(x+1)] )
=lim(x->无穷 ) (arctan(π/x)- arctan[π/(x+1)] ) /(1/x^2)
洛必达
=lim(x->无穷 ) 【 -[π/x^2]/[ 1-(π/x)^2] + [ π/(x+1)^2]/{ 1-[π/(x+1)]^2] } 】/(-2/x^3)
=lim(x->无穷 ) 【 -π/(x^2-π^2) + π/[ (x+1)^2-π^2] 】/(-2/x^3)
=(π/2)lim(x->无穷 ) x^3. ( [ (x+1)^2-π^2] -(x^2-π^2) )/【(x^2-π^2)[(x+1)^2-π^2]】
=(π/2)lim(x->无穷 ) x^3. ( 2x+1 ) /【(x^2-π^2)[(x+1)^2-π^2]】
=(π/2)lim(x->无穷 ) ( 2+ 1/x ) /【(1-π^2/x^2)[(1+1/x)^2-π^2/x^2]】
=(π/2) . { (2+0)/[(1-0)(1-0)] }
=π