Exercise · ZW-TYPES-OF-DISCONTINUITY-001 · ★☆☆ Basic
What type of discontinuity does \(f(x)=\dfrac{x^{2}-1}{x-1}\) have at \(x=1\)?
Choose one option.
Show answer
- A. Removable
Explanation
The one-sided limits exist and are equal (the limit exists), but f(a) is undefined: it is a removable discontinuity. The hole can be "filled" by defining f(a) equal to the limit.
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-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?
- ZW-TYPES-OF-DISCONTINUITY-009 What type of discontinuity does f(x)=tan x have at x=(π)/(2)?