Exercise · ZW-ONE-SIDED-LIMITS-001 · ★☆☆ Basic
For \(\lim_{x\to a} f(x)\) to exist it is necessary and sufficient that...
Choose one option.
Show answer
- B. both one-sided limits exist and are equal
Explanation
The limit exists exactly when the left-hand and right-hand limits both exist and agree. Nothing is required about f(a).
Review the concept: One-sided limits
The left-hand limit uses only values \(x<a\) and the right-hand limit only values \(x>a\). The limit of \(f\) at \(a\) exists if and only if both one-sided limits exist and are equal.
\[\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\]
More practice problems: one-sided limits
- ZW-ONE-SIDED-LIMITS-002 Compute lim _x→ 0^- (|x|)/(x).
- ZW-ONE-SIDED-LIMITS-003 Compute lim _x→ 0^+ (|x|)/(x).
- ZW-ONE-SIDED-LIMITS-004 Compute lim _x→ 2^+ (1)/(x - 2).
- ZW-ONE-SIDED-LIMITS-005 Compute lim _x→ 2^- (1)/(x - 2).
- ZW-ONE-SIDED-LIMITS-006 Compute lim _x→ 0^- (1)/(x^2).
- ZW-ONE-SIDED-LIMITS-007 Compute lim _x→ 1^-(2x+1).
- ZW-ONE-SIDED-LIMITS-008 Let f(x)=x+1 if x<2 and f(x)=x^2-1 if x≥ 2. Compute lim _x→ 2 f(x).
- ZW-ONE-SIDED-LIMITS-009 If lim _x→ a^-f(x)=2 and lim _x→ a^+f(x)=5, does lim _x→ af(x) exist?