Mathematics

Mathematics

Problem

For a function ff, the following are known: lim⁡x→2−f(x)=3,lim⁡x→2+f(x)=5,f(2)=4.\displaystyle\lim_{x\to 2^-}f(x)=3,\qquad \lim_{x\to 2^+}f(x)=5,\qquad f(2)=4. Identify the type of discontinuity at x=2x=2. a) Infinite discontinuity b) Jump discontinuity c) Removable discontinuity d) No discontinuity
Intuitive explanation

The key to classifying discontinuities is checking whether the left and right limits exist and agree. Here, both one-sided limits exist but are different (3≠53 \neq 5), which means the function "jumps" from one value to another as you cross x=2x=2. This is a jump discontinuity—the most recognizable type because the function has a finite break, not an infinite spike or wild oscillation.

Intuitive explanation

The key to classifying discontinuities is checking whether the left and right limits exist and agree. Here, both one-sided limits exist but are different (3≠53 \neq 5), which means the function "jumps" from one value to another as you cross x=2x=2. This is a jump discontinuity—the most recognizable type because the function has a finite break, not an infinite spike or wild oscillation.

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