Applied Calculus MA25C01 | ✅ Continuity Problem 1 : Find a and b

▶ Tutorial Video: Vector Spaces-Example 8

📘 Continuity of Piecewise Function

Problem:
Find the values of a and b such that the function is continuous for all x.

f(x) = {
  ax + 2b,    x ≤ 0
  x² + 3a − b,    0 < x ≤ 2
  3x − 5,    x > 2
}

🧠 Concept:

For continuity at a point x = a:

lim (x → a⁻) f(x) = lim (x → a⁺) f(x) = f(a)

🔹 At x = 0

Left limit: lim (x→0⁻) (ax + 2b) = a(0)+2b=2b
Right limit: lim (x→0⁺) (x² + 3a − b) =0^2+3a-b= 3a − b
f(0) =a(0)+2b= 2b

Continuity condition:

2b = 3a − b
⇒ 3b = 3a
⇒ a − b = 0   ...(1)

🔹 At x = 2

Left limit: lim (x→2⁻) (x² + 3a − b) = 4 + 3a − b
Right limit: lim (x→2⁺) (3x − 5) = 1
f(2) = 4 + 3a − b

Continuity condition:

4 + 3a − b = 1
⇒ 3a − b = −3   ...(2)

📊 Solving Equations

From (1): a − b = 0
From (2): 3a − b = −3

Solving, we get:
a = -3/2
b = -3/2

✅ Final Answer: a = -3/2 , b = -3/2

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✔️ Practice more problems to master continuity ✔️