The value of 'a' for which `f(x)= {((sin^2 ax)/(x^2)", " x ne 0 ),(1", "x=1):}`
is continuous at x=0 , is

Options

A. `pm 1`
B. `pm 2`
C. 0
D. `pm 3`

Answer

To find the value of 'a' for which the function

\[ f(x) = \begin{cases} \frac{\sin^2(ax)}{x^2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \]

is continuous at \(x = 0\), we need to ensure that the limit of \(f(x)\) as \(x\) approaches \(0\) equals \(f(0)\).

### Step 1: Find the limit of \(f(x)\) as \(x\) approaches \(0\)

We will calculate the right-hand limit:

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin^2(ax)}{x^2} \]

### Step 2: Substitute \(x\) with \(h\)

Let \(x = h\) where \(h \to 0\):

\[ \lim_{h \to 0} \frac{\sin^2(ah)}{h^2} \]

### Step 3: Use the limit property of \(\sin\)

We know that

\[ \lim_{h \to 0} \frac{\sin(ah)}{ah} = 1 \]

Thus, we can rewrite the limit:

\[ \lim_{h \to 0} \frac{\sin^2(ah)}{h^2} = \lim_{h \to 0} \left(\frac{\sin(ah)}{ah}\right)^2 \cdot a^2 \]

### Step 4: Apply the limit

Now applying the limit:

\[ = a^2 \cdot \lim_{h \to 0} \left(\frac{\sin(ah)}{ah}\right)^2 = a^2 \cdot 1^2 = a^2 \]

### Step 5: Set the limit equal to \(f(0)\)

Since \(f(0) = 1\), we set the limit equal to this value:

\[ a^2 = 1 \]

### Step 6: Solve for \(a\)

Taking the square root of both sides gives:

\[ a = \pm 1 \]

### Conclusion

The values of \(a\) for which the function \(f(x)\) is continuous at \(x = 0\) are:

\[ a = 1 \quad \text{or} \quad a = -1 \]

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