If lim_{x->a} f(x) = L and lim_{x->a} g(x) = M, what is lim_{x->a} f(x) g(x)?

If lim_{x->a} f(x) = L and lim_{x->a} g(x) = M, what is lim_{x->a} f(x) g(x)?

• A. L+M
• B. DNE
• C. LM
• D. ∞

Answer: (C)

When two functions have finite limits as x approaches a, the limit of their product equals the product of those limits. Here, f(x) → L and g(x) → M as x → a, so the product f(x)g(x) → L·M. This follows from the limit law for products, which applies because both limits exist and are finite. Therefore the limit is LM. The other options would require different behaviors (adding would give L+M; a nonexistence or infinity would occur if one of the limits didn’t exist or the product diverged), but with finite limits, the product limit is the finite value LM.

When two functions have finite limits as x approaches a, the limit of their product equals the product of those limits. Here, f(x) → L and g(x) → M as x → a, so the product f(x)g(x) → L·M. This follows from the limit law for products, which applies because both limits exist and are finite. Therefore the limit is LM. The other options would require different behaviors (adding would give L+M; a nonexistence or infinity would occur if one of the limits didn’t exist or the product diverged), but with finite limits, the product limit is the finite value LM.