Exercise · ZW-ONE-SIDED-LIMITS-009 · ★☆☆ Basic
If \(\lim_{x\to a^-}f(x)=2\) and \(\lim_{x\to a^+}f(x)=5\), does \(\lim_{x\to a}f(x)\) exist?
Choose one option.
Show answer
- B. No, because the one-sided limits differ
Explanation
The two-sided limit exists only if both one-sided limits agree; here 2 and 5 differ, so it does not exist.
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-001 For lim _x→ a f(x) to exist it is necessary and sufficient that...
- 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).