Exercise · ZW-TYPES-OF-DISCONTINUITY-010 · ★★☆ Intermediate
What type of discontinuity does \(f(x)=\dfrac{\sin 3x}{x}\) have at \(x=0\)?
Choose one option.
Show answer
- D. Removable
Explanation
Since \(\dfrac{\sin 3x}{3x}\to 1\), the limit at \(0\) is \(3\). Because \(f(0)\) is undefined, the discontinuity is removable.
Review the concept: Types of discontinuity
A function is discontinuous at \(a\) when some continuity condition fails. It can be removable, a jump, or infinite.
\[\lim_{x\to a^-}f(x)\neq\lim_{x\to a^+}f(x)\ \Rightarrow\ \text{jump discontinuity}\]
More practice problems: types of discontinuity
- ZW-TYPES-OF-DISCONTINUITY-001 What type of discontinuity does f(x)=(x^2-1)/(x-1) have at x=1?
- ZW-TYPES-OF-DISCONTINUITY-002 What type of discontinuity does f(x)=(|x|)/(x) have at x=0?
- ZW-TYPES-OF-DISCONTINUITY-003 What type of discontinuity does f(x)=(1)/(x-2) have at x=2?
- ZW-TYPES-OF-DISCONTINUITY-004 What type of discontinuity does f(x)=(sin x)/(x) have at x=0?
- ZW-TYPES-OF-DISCONTINUITY-005 What type of discontinuity does f(x)=(1)/(x^2) have at x=0?
- ZW-TYPES-OF-DISCONTINUITY-006 A function has lim _x→ 2f(x)=5 but f(2)=3. What type of discontinuity does it have at x=2?
- ZW-TYPES-OF-DISCONTINUITY-007 What type of discontinuity does f(x)=(x^2-4)/(x-2) have at x=2?
- ZW-TYPES-OF-DISCONTINUITY-008 Let f(x)=begincasesx+1 & x<0\\ x^2+2 & x≥ 0endcases. What type of discontinuity does it have at x=0?