Exercise · ZW-LIMIT-OF-A-FUNCTION-017 · ★★☆ Intermediate
Which pair of sequences tending to \(0\) proves that \(\sin(1/x)\) has no limit at \(0\)?
Choose one option.
Show answer
- C. \(x_n=\dfrac1{n\pi}\) and \(y_n=\dfrac1{\pi/2+2n\pi}\)
Explanation
Along \(x_n\) the value is \(\sin(n\pi)=0\) and along \(y_n\) it is \(\sin(\pi/2+2n\pi)=1\): two different limits. In the other pairs both sequences give the same value (\(0\) or \(1\)), so nothing follows.
Review the concept: Limit of a function
The limit of \(f\) as \(x\) approaches \(a\) is \(L\) if \(f(x)\) gets as close to \(L\) as we like when \(x\) is taken sufficiently close to \(a\), regardless of what happens at \(x=a\).
\[\lim_{x\to a} f(x)=L\]
More practice problems: limit of a function
- ZW-LIMIT-OF-A-FUNCTION-001 If lim _x→ 2 f(x)=5, what can be stated for sure?
- ZW-LIMIT-OF-A-FUNCTION-002 Which statement about the limit of f at a is correct?
- ZW-LIMIT-OF-A-FUNCTION-003 What does lim _x→ a f(x)=L mean?
- ZW-LIMIT-OF-A-FUNCTION-004 What is lim _x→ 3 2x?
- ZW-LIMIT-OF-A-FUNCTION-005 What is lim _x→ 2(x^2-4)/(x-2)?
- ZW-LIMIT-OF-A-FUNCTION-006 What is lim _x→ 0(sin x)/(x)?
- ZW-LIMIT-OF-A-FUNCTION-007 If f(x)=1 for x<0 and f(x)=2 for x≥ 0, what is lim _x→ 0 f(x)?
- ZW-LIMIT-OF-A-FUNCTION-008 What is lim _x→ 1(x^3-1)/(x-1)?