Exercise · ZW-ONE-SIDED-LIMITS-006 · ★★☆ Intermediate
Compute \(\lim_{x\to 0^-}\,\frac{1}{x^{2}}\).
Choose one option.
Show answer
- A. \(\infty\)
Explanation
Since \(x^{2}\) is positive and approaches 0, the quotient \(1/x^{2}\) grows without bound on both sides: the limit is \(\infty\).
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-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?