Let f(x)=x^2-4x+5, x<3 2x-1, x≥ 3. Determine whether f is continuous at x=3.

Let $f (x) = {x^{2} - 4 x + 5, 2 x - 1, x < 3 x \geq 3$ . Determine whether $f$ is continuous at $x = 3$ .

medium 2 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

Continuous at x = 3

The correct answer is $Continuous at x = 3$ — Because the left‑hand and right‑hand limits are not equal, the limit does not exist; therefore $f$ is not continuous at $x = 3$ .

Step-by-step solution

Find the left‑hand limit: $x \to 3^{-} lim f (x) = 3^{2} - 4 \cdot 3 + 5 = 9 - 12 + 5 = 2$ .
Find the right‑hand limit: $x \to 3^{+} lim f (x) = 2 \cdot 3 - 1 = 6 - 1 = 5$ .
Compute the function value at $x = 3$ : $f (3) = 2 \cdot 3 - 1 = 5$ (since $x = 3$ falls in the second piece).
Compare the limits: the left‑hand limit is $2$ while the right‑hand limit (and the function value) is $5$ .
Because the left‑hand and right‑hand limits are not equal, the limit does not exist; therefore $f$ is not continuous at $x = 3$ .

Common mistake: Considering only one side of the limit and ignoring the other side.

Concept tested: Continuity of piecewise functions

Published 9 September 2026 · How this solution was prepared and checked

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Let f(x)={x2−4x+5,2x−1,​x<3x≥3​. Determine whether f is continuous at x=3.

Step-by-step solution

Let $f (x) = {x^{2} - 4 x + 5, 2 x - 1, x < 3 x \geq 3$ . Determine whether $f$ is continuous at $x = 3$ .