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. L + M
• C. LM
• D. 0

Answer: (A)

The limit operator is linear, so the limit of a difference equals the difference of the limits. If f(x) approaches L and g(x) approaches M as x approaches a, then f(x) − g(x) will approach L − M. This follows from taking limits term by term: lim (f(x) − g(x)) = lim f(x) − lim g(x) = L − M. Intuitively, near a, f(x) stays close to L and g(x) stays close to M, so their difference stays close to L − M. The other options would correspond to different operations: adding would give L + M, multiplying would give LM, and zero would only occur if L and M happen to be equal, which isn’t guaranteed.

The limit operator is linear, so the limit of a difference equals the difference of the limits. If f(x) approaches L and g(x) approaches M as x approaches a, then f(x) − g(x) will approach L − M. This follows from taking limits term by term: lim (f(x) − g(x)) = lim f(x) − lim g(x) = L − M.

Intuitively, near a, f(x) stays close to L and g(x) stays close to M, so their difference stays close to L − M. The other options would correspond to different operations: adding would give L + M, multiplying would give LM, and zero would only occur if L and M happen to be equal, which isn’t guaranteed.