Exercise · ZW-SQUEEZE-THEOREM-001 · ★★☆ Intermediate
What is \(\lim_{x\to 0} x^{2}\sin\!\left(\dfrac{1}{x}\right)\)?
Choose one option.
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- A. 0
Explanation
Since \(-1\le \sin(1/x)\le 1\), we have \(-x^{2}\le x^{2}\sin(1/x)\le x^{2}\). Both bounds tend to 0 as x → 0, so by the squeeze theorem the limit is 0.
Review the concept: Squeeze theorem
If \(g(x)\le f(x)\le h(x)\) near \(a\) and \(g\) and \(h\) both have limit \(L\) at \(a\), then \(f\) also has limit \(L\) at \(a\).
\[g\le f\le h,\ \lim_{x\to a}g=\lim_{x\to a}h=L\ \Rightarrow\ \lim_{x\to a}f=L\]
More practice problems: squeeze theorem
- ZW-SQUEEZE-THEOREM-002 If 1-x^2≤ f(x)≤ 1+x^2 for all x near 0, what is lim _x→ 0 f(x)?
- ZW-SQUEEZE-THEOREM-003 What hypothesis does the squeeze theorem need to conclude that lim _x→ a f(x)=L?
- ZW-SQUEEZE-THEOREM-004 What is lim _x→ 0 xcos \!((1)/(x))?
- ZW-SQUEEZE-THEOREM-005 If g(x)≤ f(x)≤ h(x) near a, lim _x→ a g=5 and lim _x→ a h=5, what is lim _x→ a f(x)?
- ZW-SQUEEZE-THEOREM-006 What is lim _x→∞(sin x)/(x)?
- ZW-SQUEEZE-THEOREM-007 Let lfloor yrfloor be the greatest integer less than or equal to y. What is lim _x→ 0^+ xlfloor 1/x…
- ZW-SQUEEZE-THEOREM-008 Let f(x)=x^2sin (1/x) for xne 0 and f(0)=0. What is f'(0)=lim _h→ 0(f(h)-f(0))/(h)?
- ZW-SQUEEZE-THEOREM-009 What other name does the squeeze theorem have?