Question

Why x's sign is negative after being square rooted ???
Here is the equation :
Lim √(x^2 +1)/x , as x approaches -∞

Answer 1

The square-root did not become negative, it's just we have rejected the positive root.

In the original expression, sqrt(x^2+1)/x, we know that numerator is positive, and denominator is negative, so the limit must be negative, if it exists.

I suppose you squared both the numerator and denominator before taking the limit, which turns both into positive. When you take the square-root, you have to check which one is valid, the positive, or the negative.

As we said earlier, the limit, if it exists, is negative. Therefore we must reject the positive square-root, leaving the limit as negative.