We have to find lim x--> inf, [1/(x - sqrt(x^2 + 2x))].

[1/(x - sqrt(x^2 + 2x))]

=> (x + sqrt(x^2 + 2x)/ (x - sqrt(x^2 + 2x)* (x + sqrt(x^2 + 2x)

=> (x + sqrt(x^2 + 2x)/ x^2 - x^2 - 2x

=> (x + sqrt(x^2 + 2x)/-2x

...

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We have to find lim x--> inf, [1/(x - sqrt(x^2 + 2x))].

[1/(x - sqrt(x^2 + 2x))]

=> (x + sqrt(x^2 + 2x)/ (x - sqrt(x^2 + 2x)* (x + sqrt(x^2 + 2x)

=> (x + sqrt(x^2 + 2x)/ x^2 - x^2 - 2x

=> (x + sqrt(x^2 + 2x)/-2x

=> (x/-2x) – sqrt (x^2/4x^2 + 2x/4x^2)

=> -1/2 – sqrt (1/4 + 1/2x)

The required limit is now:

lim x--> +inf [-1/2 – sqrt (1/4 + 1/2x)]

substitute x = + inf, 1/x = 0

=> -1/2 – sqrt (1/4)

=> -1/2 – 1/2

=> -1

**The required result of the limit is -1.**