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# evaluate the limitevaluate the

evaluate the limit

1.lim_x->(拍/4)- (tanx)^(tan2x)

2.lim_x->無限大 {(希格瑪_j:1~n) (n^2-j^2)^1/2 over n^2}

Update:

第一題的第3~4行

第二題的第3行

看不懂

請螞蟻雄兵大大解釋一下

### 2 Answers

Rating

- Scharze spaceLv 71 decade agoFavorite Answer
1.lim(x->π/4) (tanx)^(tan2x)

=lim(x->π/4)e^[(tan2x)ln(tanx)]

=e^lim(x->π/4)[(tan2x)ln(tanx)][ e^x is continuous]

=e^lim(x->π/4)[sec^2x/tanx]/[-(tan2x)^(-1)sec^22x*2](0/0 上下微分)

=e^lim(x->π/4)[(-2sec^2x)tan2x/tanx]/[sec^2(2x)]

=e^lim(x->π/4)-4(sec^2xcosxcos2x(cosx)/(sinx)

=e^0

=1

2 lim(n->∞)Σ(j=1~n)(1/n^2)(√(n^2-j^2))

=lim(n->∞)(1/n)Σ√(n^2-j^2)/n

=lim(n->∞)(1/n)[Σ√(1-(j/n)^2)]

=∫_[0,1]√(1-x^2)dx

=π/4

=

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