Question

If the function $f(x) = \begin{cases} ax+1, & x \leq 3 \\ bx+3, & x > 3 \end{cases}$ is continuous at $x = 3$, then find the values of $a$ and $b$.

Hint:

For piecewise functions → apply $\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = f(c)$ for continuity.

Answer 1

Condition for continuity at $x=3$: \[ \lim_{x\to 3^-} f(x) = f(3) = \lim_{x\to 3^+} f(x) \]

Step 1: Left-hand limit ($x \leq 3$). \[ \lim_{x\to 3^-} f(x) = a(3) + 1 = 3a + 1 \]

Step 2: Value at $x=3$. Since $x=3$ is included in first case: \[ f(3) = 3a + 1 \]

Step 3: Right-hand limit ($x > 3$). \[ \lim_{x\to 3^+} f(x) = b(3) + 3 = 3b + 3 \]

Step 4: Apply continuity condition. \[ 3a + 1 = 3b + 3 \] \[ 3a - 3b = 2 \implies a - b = \tfrac{2}{3} \] Conclusion: The values of $a$ and $b$ are related by: \[ \boxed{a - b = \tfrac{2}{3}} \]