If lim f = L is finite and lim g = ∞ as x approaches a, what is lim f(x)/g(x)?

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Multiple Choice

If lim f = L is finite and lim g = ∞ as x approaches a, what is lim f(x)/g(x)?

Explanation:
When one function approaches a finite value and the other grows without bound, their ratio tends to zero. Think of f(x) approaching L and g(x) growing to infinity. Write f(x) = L + [f(x) − L], so f(x)/g(x) = L/g(x) + [f(x) − L]/g(x). As x approaches a, g(x) → ∞, so L/g(x) → 0. Also f(x) − L → 0 while g(x) → ∞, making [f(x) − L]/g(x) → 0. The sum of two zeros is 0, so f(x)/g(x) → 0. Therefore the limit is 0.

When one function approaches a finite value and the other grows without bound, their ratio tends to zero. Think of f(x) approaching L and g(x) growing to infinity. Write f(x) = L + [f(x) − L], so

f(x)/g(x) = L/g(x) + [f(x) − L]/g(x).

As x approaches a, g(x) → ∞, so L/g(x) → 0. Also f(x) − L → 0 while g(x) → ∞, making [f(x) − L]/g(x) → 0. The sum of two zeros is 0, so f(x)/g(x) → 0. Therefore the limit is 0.

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