Evaluate lim_{x->0} (√(x+1) - 1)/x.

Evaluate lim_{x->0} (√(x+1) - 1)/x.

• A. 1
• B. 0
• C. -1/2
• D. 1/2

Answer: (D)

When you see a square root difference like √(x+1) − 1 over x, the 0/0 form hints at rationalizing the numerator to simplify. Multiply top and bottom by √(x+1) + 1. The numerator becomes (√(x+1))^2 − 1^2 = x, so the expression simplifies to x / [ x(√(x+1) + 1) ] = 1 / (√(x+1) + 1). As x → 0, √(x+1) → 1, giving 1 / (1 + 1) = 1/2. This limit also matches the derivative of √t at t = 1, which is 1/(2√1) = 1/2.

When you see a square root difference like √(x+1) − 1 over x, the 0/0 form hints at rationalizing the numerator to simplify. Multiply top and bottom by √(x+1) + 1. The numerator becomes (√(x+1))^2 − 1^2 = x, so the expression simplifies to x / [ x(√(x+1) + 1) ] = 1 / (√(x+1) + 1). As x → 0, √(x+1) → 1, giving 1 / (1 + 1) = 1/2. This limit also matches the derivative of √t at t = 1, which is 1/(2√1) = 1/2.