Evaluate lim x→∞ sqrt(x^2 + x) − x.

Evaluate lim x→∞ sqrt(x^2 + x) − x.

• A. 0
• B. -1/2
• C. 1/2
• D. 1

Answer: (C)

This limit hinges on using the conjugate to remove the square root and reveal a finite result when the two large terms would otherwise cancel each other. Multiply the expression by the conjugate over itself: sqrt(x^2+x) − x = [ (x^2+x) − x^2 ] / [ sqrt(x^2+x) + x ] = x / [ sqrt(x^2+x) + x ]. For x > 0, factor x from the square root: sqrt(x^2+x) = x sqrt(1 + 1/x). Then the expression becomes 1 / [ sqrt(1 + 1/x) + 1 ]. As x → ∞, 1/x → 0, so sqrt(1 + 1/x) → 1. Therefore the limit is 1 / (1 + 1) = 1/2.

This limit hinges on using the conjugate to remove the square root and reveal a finite result when the two large terms would otherwise cancel each other.

Multiply the expression by the conjugate over itself:

sqrt(x^2+x) − x = [ (x^2+x) − x^2 ] / [ sqrt(x^2+x) + x ] = x / [ sqrt(x^2+x) + x ].

For x > 0, factor x from the square root: sqrt(x^2+x) = x sqrt(1 + 1/x). Then the expression becomes

1 / [ sqrt(1 + 1/x) + 1 ].

As x → ∞, 1/x → 0, so sqrt(1 + 1/x) → 1. Therefore the limit is 1 / (1 + 1) = 1/2.