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Lim (√(x + a)(x + b) - x
x→~

Lim (√x² + xb + xa + ab) - x)
x→~

Lim x² + xb + xa + ab - x²
x→~ .. / (√x² + xb + xa + ab) + x

Lim (xb + xa + ab)
x→~ .. / √x² (1 + b/x + a/x + ab/x²) + x

Lim x (b + a + ab/x)
x→~ .. / x (√1 + b/x + a/x + ab/x²) + x

Lim (b + a + ab/x)
x→~ .. / (√1 + b/x + a/x + ab/x²) + 1

= b + a + 0 / (√1 + 0 + 0 + 0) + 1

= b + a / (√1) + 1

= b + a / 1 + 1

= b + a / 2

= a + b / 2