1.4 Q-3

Question Statement

Discuss the continuity of the function

$$f(x) = \begin{cases} 3x & \text{if } x \leq -2, x^2 - 1 & \text{if } -2 < x < 2, 3 & \text{if } x \geq 2 \end{cases}$$

at $x = 2$ and $x = -2$.

Background and Explanation

To determine the continuity of a piecewise function at a specific point $x = c$, we must verify three conditions:

1. $f(c)$ is defined.
2. The left-hand limit ($\lim_{x \to c^-} f(x)$) and the right-hand limit ($\lim_{x \to c^+} f(x)$) exist.
3. $f(c)$, $\text{L.H.L}$, and $\text{R.H.L}$ are equal.

If any condition is not satisfied, the function is discontinuous at $x = c$.

Solution

Continuity at $x = 2$

Step 1: Evaluate $f(2)$

From the definition of the function for $x \geq 2$:

$$f(2) = 3$$

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

For $x < 2$, $f(x) = x^2 - 1$. Substituting $x \to 2^-$:

$$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 - 1) = 2^2 - 1 = 4 - 1 = 3$$

Step 3: Compute Right-Hand Limit (R.H.L)

For $x \geq 2$, $f(x) = 3$. Substituting $x \to 2^+$:

$$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} 3 = 3$$

Step 4: Verify Continuity

Since $f(2) = 3$, $\text{L.H.L} = 3$, and $\text{R.H.L} = 3$, the function is continuous at $x = 2$.

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Continuity at $x = -2$

Step 1: Evaluate $f(-2)$

From the definition of the function for $x \leq -2$:

$$f(-2) = 3(-2) = -6$$

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

For $x < -2$, $f(x) = 3x$. Substituting $x \to -2^-$:

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

Step 3: Compute Right-Hand Limit (R.H.L)

For $-2 < x < 2$, $f(x) = x^2 - 1$. Substituting $x \to -2^+$:

$$\lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (x^2 - 1) = (-2)^2 - 1 = 4 - 1 = 3$$

Step 4: Verify Continuity

Here:

• $f(-2) = -6$,
• $\text{L.H.L} = -6$,
• $\text{R.H.L} = 3$.

Since $\text{L.H.L} \neq \text{R.H.L}$ and $f(-2) \neq \text{L.H.L}$, the function is discontinuous at $x = -2$.

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Key Formulas or Methods Used

1. Definition of Continuity:
   A function $f(x)$ is continuous at $x = c$ if:

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

2. Limit Calculation for Piecewise Functions:
   • For each piece, identify the function applicable for the specific range.
   • Substitute the value of $x$ into the corresponding limit expression.

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Summary of Steps

1. Continuity at $x = 2$:

   • $f(2) = 3$, $\text{L.H.L} = 3$, $\text{R.H.L} = 3$.
   • Conclusion: Continuous at $x = 2$.

2. Continuity at $x = -2$:

   • $f(-2) = -6$, $\text{L.H.L} = -6$, $\text{R.H.L} = 3$.
   • Conclusion: Discontinuous at $x = -2$.