1.4 Q-5

Question Statement

Find the values of $m$ and $n$ such that the following functions are continuous at the specified points:

1. $$$$

mx & \text{if } x < 3
n & \text{if } x = 3
-2x + 9 & \text{if } x > 3 \end{cases}$$

2. $$$$

mx & \text{if } x < 4
x^2 & \text{if } x \geq 4 \end{cases}$$

Background and Explanation

For a function $f(x)$ to be continuous at a point $x = a$, it must satisfy:

1. $f(a)$ exists.
2. $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.
3. The value of $f(a)$ equals the common limit.

The above properties imply that the left-hand limit (L.H.L) and right-hand limit (R.H.L) must be equal and match the function’s value at $x = a$.

Solution

Part 1: Continuity at $x = 3$

Step 1: Function Value at $x = 3$

The value of $f(3)$ is given as $n$.

$$f(3) = n \tag{1}$$

Step 2: Evaluate Right-Hand Limit (R.H.L)

For $x > 3$, $f(x) = -2x + 9$.

$$\lim_{x \to 3^+} f(x) = -2(3) + 9 = -6 + 9 = 3 \tag{2}$$

Step 3: Evaluate Left-Hand Limit (L.H.L)

For $x < 3$, $f(x) = mx$.

$$\lim_{x \to 3^-} f(x) = m(3) = 3m \tag{3}$$

Step 4: Apply Continuity Condition

Since $f(x)$ is continuous at $x = 3$:

$$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)$$

Substitute the results:

$$3m = 3 \quad \text{and} \quad n = 3$$

Solving for $m$:

$$m = 1$$

Final Answer (Part 1)

$$m = 1, \quad n = 3$$

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Part 2: Continuity at $x = 4$

Step 1: Function Value at $x = 4$

For $x \geq 4$, $f(x) = x^2$.

$$f(4) = 4^2 = 16 \tag{4}$$

Step 2: Evaluate Left-Hand Limit (L.H.L)

For $x < 4$, $f(x) = mx$.

$$\lim_{x \to 4^-} f(x) = m(4) = 4m \tag{5}$$

Step 3: Apply Continuity Condition

Since $f(x)$ is continuous at $x = 4$:

$$\lim_{x \to 4^-} f(x) = f(4)$$

Substitute the results:

$$4m = 16$$

Solving for $m$:

$$m = \frac{16}{4} = 4$$

Final Answer (Part 2)

$$m = 4$$

Key Formulas or Methods Used

1. Continuity Condition:
   $f(a)$, $\lim_{x \to a^-} f(x)$, and $\lim_{x \to a^+} f(x)$ must all be equal at $x = a$.

2. Limit Evaluation for Piecewise Functions:
   Evaluate the limit from each side using the corresponding part of the piecewise function.

Summary of Steps

Part 1:

1. Evaluate $f(3) = n$, $\lim_{x \to 3^-} f(x) = 3m$, and $\lim_{x \to 3^+} f(x) = 3$.
2. Use continuity conditions to find $m = 1$ and $n = 3$.

Part 2:

1. Evaluate $f(4) = 16$ and $\lim_{x \to 4^-} f(x) = 4m$.
2. Use continuity conditions to find $m = 4$.