Exercise · ZW-TYPES-OF-DISCONTINUITY-006 · ★☆☆ Basic
A function has \(\lim_{x\to 2}f(x)=5\) but \(f(2)=3\). What type of discontinuity does it have at \(x=2\)?
Choose one option.
Show answer
- D. Removable
Explanation
The limit exists (it equals \(5\)) but does not match \(f(2)=3\). Redefining \(f(2)\) as \(5\) removes the discontinuity: it 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-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?
- ZW-TYPES-OF-DISCONTINUITY-009 What type of discontinuity does f(x)=tan x have at x=(π)/(2)?
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