Find  
 
 
 
\[\displaystyle\lim_{x\to 3}\dfrac{9-x^2}{\cos\left(\dfrac\pi2x\right)}\].


To solve the limit \(\displaystyle\lim_{x\to 3}\dfrac{9-x^2}{\cos\left(\dfrac{\pi}{2}x\right)}\), let's analyze the behavior of both the numerator and the denominator as \(x\) approaches 3.

### Step 1: Analyze the numerator
The numerator is \(9 - x^2\). As \(x\) approaches 3:
\[
9 - x^2 = 9 - 9 = 0
\]

### Step 2: Analyze the denominator
The denominator is \(\cos\left(\dfrac{\pi}{2}x\right)\). As \(x\) approaches 3:
\[
\cos\left(\dfrac{\pi}{2} \times 3\right) = \cos\left(\dfrac{3\pi}{2}\right)
\]
The value of \(\cos\left(\dfrac{3\pi}{2}\right)\) is 0. Thus, the denominator also approaches 0 as \(x\) approaches 3.

### Step 3: Apply L'Hôpital's Rule
Since both the numerator and denominator approach 0, we have a \(\frac{0}{0}\) indeterminate form, so we can apply L'Hôpital's Rule. According to L'Hôpital's Rule, we differentiate the numerator and the denominator with respect to \(x\) and then take the limit.

Differentiate the numerator:
\[
\frac{d}{dx}(9 - x^2) = -2x
\]

Differentiate the denominator:
\[
\frac{d}{dx}\left(\cos\left(\dfrac{\pi}{2}x\right)\right) = -\sin\left(\dfrac{\pi}{2}x\right) \cdot \frac{\pi}{2}
\]

So, the limit now becomes:
\[
\lim_{x \to 3} \dfrac{-2x}{-\sin\left(\dfrac{\pi}{2}x\right) \cdot \frac{\pi}{2}} = \lim_{x \to 3} \dfrac{-2x}{-\frac{\pi}{2} \sin\left(\dfrac{\pi}{2}x\right)} 
\]

Simplifying this expression:
\[
\lim_{x \to 3} \dfrac{2x}{\frac{\pi}{2} \sin\left(\dfrac{\pi}{2}x\right)} = \lim_{x \to 3} \dfrac{4x}{\pi \sin\left(\dfrac{\pi}{2}x\right)}
\]

### Step 4: Substitute \(x = 3\) into the expression
Substituting \(x = 3\) into the simplified expression:
\[
\dfrac{4 \times 3}{\pi \sin\left(\dfrac{3\pi}{2}\right)} = \dfrac{12}{\pi \times (-1)} = -\dfrac{12}{\pi}
\]

### Final Answer
Therefore, the limit is:
\[
\lim_{x\to 3}\dfrac{9-x^2}{\cos\left(\dfrac{\pi}{2}x\right)} = -\dfrac{12}{\pi}
\]

Would you like further details on any of the steps? Here are some related questions you might be interested in:

1. How do we determine when to apply L'Hôpital's Rule?
2. What are the other indeterminate forms where L'Hôpital's Rule is applicable?
3. How would the limit change if the function involved \(\sin\) instead of \(\cos\)?
4. What is the significance of \(\cos\) and \(\sin\) values at specific multiples of \(\pi\)?
5. Can this limit be solved using a Taylor series expansion?

**Tip:** Always ensure that you check the indeterminate form before applying L'Hôpital's Rule.

Explore how to solve the limit problem lim (9 - x^2) / cos(pi/2 * x) as x approaches 3 with a detailed step-by-step solution using L'Hôpital's Rule. This problem involves trigonometric functions and limits, suitable for advanced high school or college students.

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