Exercise · ZW-TYPES-OF-DISCONTINUITY-008 · ★★★ Advanced
Let \(f(x)=\begin{cases}x+1 & x<0\\ x^{2}+2 & x\ge 0\end{cases}\). What type of discontinuity does it have at \(x=0\)?
Choose one option.
Show answer
- B. Jump
Explanation
The left-hand limit is \(1\) and the right-hand limit is \(2\), which differ even though \(f(0)=2\) exists. So the graph jumps: a jump discontinuity.
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-009 What type of discontinuity does f(x)=tan x have at x=(π)/(2)?
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