Exercise · ZW-TYPES-OF-DISCONTINUITY-007 · ★★☆ Intermediate
What type of discontinuity does \(f(x)=\dfrac{x^{2}-4}{x-2}\) have at \(x=2\)?
Choose one option.
Show answer
- C. Removable
Explanation
Factoring gives \(x^{2}-4=(x-2)(x+2)\), so the limit at \(2\) is \(4\). Since \(f(2)\) is undefined, the discontinuity is removable, not infinite.
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
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- 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-008 Let f(x)=begincasesx+1 & x<0\\ x^2+2 & x≥ 0endcases. What type of discontinuity does it have at x=0?
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