Evaluate the limit as x approaches infinity of x/(x + sqrt(x^2 + 1)).

Evaluate the limit as x approaches infinity of x/(x + sqrt(x^2 + 1)).

• A. 1/2
• B. 0
• C. 1
• D. 2

Answer: (A)

When a limit as x grows without bound involves sums of terms that grow with x, simplify by dividing by the largest term to compare their growth directly. Divide numerator and denominator by x: x/(x + sqrt(x^2+1)) = 1 / (1 + sqrt(1 + 1/x^2)). As x → ∞, 1/x^2 → 0, so sqrt(1 + 1/x^2) → 1, and the expression tends to 1 / (1 + 1) = 1/2. Intuition: both parts of the denominator behave like x for large x, so the ratio behaves like x/(2x) = 1/2.

When a limit as x grows without bound involves sums of terms that grow with x, simplify by dividing by the largest term to compare their growth directly. Divide numerator and denominator by x: x/(x + sqrt(x^2+1)) = 1 / (1 + sqrt(1 + 1/x^2)). As x → ∞, 1/x^2 → 0, so sqrt(1 + 1/x^2) → 1, and the expression tends to 1 / (1 + 1) = 1/2. Intuition: both parts of the denominator behave like x for large x, so the ratio behaves like x/(2x) = 1/2.