Exercise · ZW-ONE-SIDED-LIMITS-010 · ★★☆ Intermediate
Compute \(\lim_{x\to 0^+}\,\frac{\sin x}{|x|}\).
Choose one option.
Show answer
- A. \(1\)
Explanation
For \(x>0\), \(|x|=x\), so the quotient is \(\dfrac{\sin x}{x}\), which tends to \(1\).
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).