A function satisfies f(2) = 5, but lim x→ 2 f(x) = 3. What does the graph look like near x = 2?

The curve heads toward height 5, with a hole at height 3

The curve jumps from 3 up to 5 as x crosses 2

Impossible, the limit must equal the value

Physics I / Limits and continuity

Practice question · Multiple choice

A function satisfies $f(2) = 5$ , but $\lim_{x\to 2} f(x) = 3$ . What does the graph look like near $x = 2$ ?

Hints

The limit reports where the curve is heading; the value reports where the point was placed. They are independent pieces of data.
Approach from both sides along the curve, you drift toward 3. The definition $f(2)=5$ just parks a dot elsewhere.

Show the answer

C. The curve heads toward 3, with a lone dot at 5

Why

This is a removable discontinuity: the curve funnels toward 3, but someone stuck the actual point at 5. Continuity is precisely the demand that these two, the heading and the landing, agree.

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A function satisfies f(2)=5f(2) = 5f(2)=5, but lim⁡x→2f(x)=3\lim_{x\to 2} f(x) = 3limx→2​f(x)=3. What does the graph look like near x=2x = 2x=2?

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A function satisfies $f(2) = 5$ , but $\lim_{x\to 2} f(x) = 3$ . What does the graph look like near $x = 2$ ?