Çözüldü Limit - Cebirsel Özdeşlikler

https://i72.servimg.com/u/f72/19/97/10/39/limit47.png
https://cdn.eba.gov.tr/yardimcikaynaklar/2022/11/kt/12kt/mat/22.pdf
(Soru 10, Yanıtlar yok)

Çözüm - 1: (L'Hospital Kuralı ile)
lim (x → 1) { 1 / (2√x) + 1 / [ 2√(x - 1) ] } / { 2x / [ 2√(x^2 - 1) ] } =
(1 / 2)·lim (x → 1) [ 1 / √x + 1 / √(x - 1) ] / [ x / √(x^2 - 1) ] =
(1 / 2)·lim (x → 1) { [ √(x - 1) + √x ] / [ √(x - 1)·√x ] }·{ [ √(x - 1)·√(x + 1) ] / x } =
(1 / 2)·lim (x → 1) { [ √(x - 1) + √x ] / √x }·{ [ √(x + 1) ] / x } =
(1 / 2)·[ (0 + 1) / 1 ]·[ (√2) / 1 ] =
(√2) / 2.

Çözüm - 2:
lim (x → 1) (√x) / √(x^2 - 1) + lim (x → 1) √(x - 1) / √(x^2 - 1) - lim (x → 1) 1 / √(x^2 - 1) =
lim (x → 1) (√x - 1) / √(x^2 - 1) + lim (x → 1) 1 / √(x + 1) =
lim (x → 1) (√x - 1)(√x + 1) / [ (√x + 1)·√(x - 1) ] + 1 / √2 =
lim (x → 1) (x - 1) / [ (√x + 1)·√(x - 1) ] + (√2) / 2 =
lim (x → 1) [ (x - 1)^2 ]^0,5 / [ (√x + 1)·√(x - 1) ] + (√2) / 2 =
lim (x → 1) [ √(x - 1) / √(x + 1) ] + (√2) / 2 =
0 / √2 + (√2) / 2 =
0 + (√2) / 2 =
(√2) / 2.