Evaluate lim x→∞ x/(sqrt(x^2+1)).

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Multiple Choice

Evaluate lim x→∞ x/(sqrt(x^2+1)).

Explanation:
When x grows without bound, simplify by factoring x^2 inside the square root. Write sqrt(x^2+1) = |x| sqrt(1+1/x^2). For x approaching positive infinity, |x| = x, so the expression becomes x / [x sqrt(1+1/x^2)] = 1 / sqrt(1+1/x^2). As x → ∞, 1/x^2 → 0, giving 1 / sqrt(1) = 1. Therefore, the limit is 1. Note that if x → -∞, the limit would be -1 because |x| = -x in that direction.

When x grows without bound, simplify by factoring x^2 inside the square root. Write sqrt(x^2+1) = |x| sqrt(1+1/x^2). For x approaching positive infinity, |x| = x, so the expression becomes x / [x sqrt(1+1/x^2)] = 1 / sqrt(1+1/x^2). As x → ∞, 1/x^2 → 0, giving 1 / sqrt(1) = 1. Therefore, the limit is 1. Note that if x → -∞, the limit would be -1 because |x| = -x in that direction.

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