Which statement correctly expresses the formal epsilon-delta definition of lim_{x->a} f(x) = L?

Which statement correctly expresses the formal epsilon-delta definition of lim_{x->a} f(x) = L?

• A. For every epsilon > 0 there exists delta > 0 such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.
• B. For every epsilon > 0 there exists delta > 0 such that |f(x) - L| < epsilon implies 0 < |x - a| < delta.
• C. There exists a delta > 0 such that for all x with 0 < |x - a| < delta, |f(x) - L| ≤ epsilon.
• D. If x is near a, then f(x) is near L.

Answer: (A)

The statement tested is the formal epsilon-delta definition of a limit. It says that no matter how small a margin of error epsilon you choose, you can pick a neighborhood around a (delta) so that whenever x lies inside that neighborhood but is not equal to a, f(x) lands within epsilon of L. The importance lies in the exact order of quantifiers: for every epsilon > 0 there exists delta > 0 such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon. This captures the idea that you can make f(x) as close as you like to L by bringing x sufficiently close to a (without requiring x to equal a). The condition 0 < |x - a| < delta excludes x = a, which is essential for the limit concept. The other formulations either flip the implication, fix epsilon with a single delta, or state proximity in a vague way, and thus do not match the precise logical structure needed for the limit definition.

The statement tested is the formal epsilon-delta definition of a limit. It says that no matter how small a margin of error epsilon you choose, you can pick a neighborhood around a (delta) so that whenever x lies inside that neighborhood but is not equal to a, f(x) lands within epsilon of L. The importance lies in the exact order of quantifiers: for every epsilon > 0 there exists delta > 0 such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon. This captures the idea that you can make f(x) as close as you like to L by bringing x sufficiently close to a (without requiring x to equal a). The condition 0 < |x - a| < delta excludes x = a, which is essential for the limit concept. The other formulations either flip the implication, fix epsilon with a single delta, or state proximity in a vague way, and thus do not match the precise logical structure needed for the limit definition.