Let the function f(x) = x² + 9. What is lim(x→0) (√f(x) − 3)/(√(f(x) + 7) − 4) equal to?

f(x) = x² + 9. At x = 0, f(0) = 9, √f(0) = 3, √(f(0) + 7) = √16 = 4. Both numerator and denominator → 0. Rationalize: (√f − 3)/(√(f+7) − 4) × (√f + 3)(√(f+7) + 4) / ((√f + 3)(√(f+7) + 4)) = (f − 9)(√(f+7) + 4) / ((f + 7 − 16)(√f + 3)) = (f − 9)(√(f+7) + 4) / ((f − 9)(√f + 3)) = (√(f+7) + 4)/(√f + 3). At x = 0: (4 + 4)/(3 + 3) = 8/6 = 4/3.