limx→−∞2xx2+1√=

To solve the limit of the function as x approaches negative infinity, we can use the method of dividing the numerator and the denominator by the highest power of x in the denominator. Here, the highest power of x in the denominator is x^2.

Step 1: Divide every term in the function by x^2:

lim (x→-∞) (2x/x^2) / (√(x^2/x^2 + 1/x^2))

This simplifies to:

lim (x→-∞) (2/x) / (√(1 + 1/x^2))

Step 2: As x approaches negative infinity, 2/x approaches 0 and 1/x^2 also approaches 0. So, the function simplifies to:

lim (x→-∞) 0 / √1

Step 3: 0 divided by any number is 0. So, the limit of the function as x approaches negative infinity is 0.