Can you help me with limits please?!! :(

1) lim (x->∞) { (square root)(4+x-x^2) - x } / x^2
2) lim (x->∞) (square root)(x^2 +x) - (square root)(x^2 +5)

Use conjugates.

1) lim(x→∞) [√(4 + x - x^2) - x] / x^2
= lim(x→∞) [√(4 + x - x^2) - x] * [√(4 + x - x^2) + x] / {x^2 * [√(4 + x - x^2) + x]}
= lim(x→∞) [(4 + x - x^2) - x^2] / {x^2 * [√(4 + x - x^2) + x]}
= lim(x→∞) (x - x^2) / {x^2 * [√(4 + x - x^2) + x]}
= lim(x→∞) (1/x - 1) / [√(4 + x - x^2) + x]
= 0, since the numerator approaches -1, while the denominator approaches ∞.
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2) lim(x→∞) [√(x^2 + x) - √(x^2 + 5)]
= lim(x→∞) [√(x^2 + x) - √(x^2 + 5)] * [√(x^2 + x) + √(x^2 + 5)]/[√(x^2 + x) + √(x^2 + 5)]
= lim(x→∞) [(x^2 + x) - (x^2 + 5)]/[√(x^2 + x) + √(x^2 + 5)]
= lim(x→∞) (x - 5) / [√(x^2 + x) + √(x^2 + 5)]

Divide each term in the numerator and denominator by x = √(x^2):
lim(x→∞) (1 - 5/x) / [√(1 + 1/x) + √(1 + 5/x)]
= (1 - 0)/(1 + 1)
= 1/2.
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I hope this helps!