What is the sequential criterion for lim_{x->a} f(x) = L?

What is the sequential criterion for lim_{x->a} f(x) = L?

• A. For every sequence x_n -> a with x_n ≠ a, f(x_n) -> L.
• B. For some sequence x_n -> a with x_n ≠ a, f(x_n) -> L.
• C. For every sequence x_n -> a with x_n ≠ a, f(x_n) -> not L.
• D. If x_n is constant equal to a for all n, then f(x_n) -> L.

Answer: (A)

As you approach a, the values of f(x) must settle near L no matter how you approach, as long as you stay away from a. In sequential terms, that means: for every sequence x_n that converges to a with x_n ≠ a, the sequence f(x_n) converges to L. This captures all possible ways of approaching a and ensures the limit is well-defined. If you only required some sequence to converge to L, that wouldn’t guarantee the limit, because another way of approaching a could give outputs that don’t get close to L. If you required every sequence to converge to the opposite value, that would contradict having limit L in the first place. And the condition doesn’t use a constant sequence at a, since the approach must come from values near a, not from staying at a.

As you approach a, the values of f(x) must settle near L no matter how you approach, as long as you stay away from a. In sequential terms, that means: for every sequence x_n that converges to a with x_n ≠ a, the sequence f(x_n) converges to L. This captures all possible ways of approaching a and ensures the limit is well-defined.

If you only required some sequence to converge to L, that wouldn’t guarantee the limit, because another way of approaching a could give outputs that don’t get close to L. If you required every sequence to converge to the opposite value, that would contradict having limit L in the first place. And the condition doesn’t use a constant sequence at a, since the approach must come from values near a, not from staying at a.