Determine lim_{x->-∞} x^2/(x^2 + 1).

Determine lim_{x->-∞} x^2/(x^2 + 1).

• A. 0
• B. 1
• C. 1/2
• D. ∞

Answer: (B)

As x grows without bound in the negative direction, x^2 becomes very large and positive. The expression can be rewritten as x^2/(x^2+1) = 1/(1 + 1/x^2). With x^2 → ∞, we have 1/x^2 → 0, so the whole fraction tends to 1. The ratio stays below 1 for all x because the denominator is always larger than the numerator by 1. Therefore, the limit is 1.

As x grows without bound in the negative direction, x^2 becomes very large and positive. The expression can be rewritten as x^2/(x^2+1) = 1/(1 + 1/x^2). With x^2 → ∞, we have 1/x^2 → 0, so the whole fraction tends to 1. The ratio stays below 1 for all x because the denominator is always larger than the numerator by 1. Therefore, the limit is 1.