Prove limx - > 1 (x-1) / (√ X-1 = 2) with the definition of function limit Prove with the definition of function lim x->1 （x-1）/ （√x -1）=2

Prove limx - > 1 (x-1) / (√ X-1 = 2) with the definition of function limit Prove with the definition of function lim x->1 （x-1）/ （√x -1）=2

Need: | (x-1) / (√ x-1) - 2 | = | √ X-1 | = | X-1 | / (√ x + 1)|

It is proved that limx → a 1 / x = 1 / a limit holds by the definition of function limit
It is proved that LIM 1 / x = 1 / a limit holds by the definition of function limit
x→a
I know 0 < x - a < δ, so | 1 / X-1 / a < ε
|(a-x)/(ax)|=|x-a|/(|a|*|x|)< δ/(|a|*|x|)
If it's the answer I want, I can add more points

|X | > | a | - 1 | (this number can be arbitrary, as long as it is less than | x |, generally take the number closest to X and easy to find), then | (A-X) / (AX) | = | x-a | / (| a | * | x |) < δ / (| a | * | x |) < δ / (| a | * | a | - 1 |)