Question

lim x->1 (sqrt(x) - 1)/ (x - 1)

Answer 1

hint :
\((x-1)=(\sqrt x)^2-1^2 \)
now use the difference of squares \(a^2-b^2=(a+b)(a-b)\)

Answer 2

how do i do that?

Answer 3

\((x-1)=(\sqrt x)^2-1^2=(\sqrt x-1)(\sqrt x+1)\)
got this ? what gets cancelled ?

Answer 4

okay the sqrt(x) - 1 gets cancelled

Answer 5

yes, you can directly put x=1 now.

Answer 6

okay so i substitute x = 1 into sqrt(x) - 1?

Answer 7

no....that got cancelled....what remains ?

Answer 8

what remains is (x - 1)

Answer 9

but we wrote x-1 as
\((x-1)=(\sqrt x)^2-1^2=(\sqrt x-1)(\sqrt x+1)\)

Answer 10

ok

Answer 11

\(\dfrac{\sqrt x-1}{x-1}=\dfrac{\sqrt{x}-1}{(\sqrt x-1 )(\sqrt x +1)}=\dfrac{1}{\sqrt x+1}\)
got this ?
now put x=1

Answer 12

ok i get 1/2

Answer 13

i get the same, its correct.

Answer 14

ok i get 1/2

Answer 15

yes, 1/2 is correct.