Evaluate the following limits :
`Lim_(theta to pi/2)(cot theta)/(pi/2 - theta )`

Answer

To evaluate the limit

\[ \lim_{\theta \to \frac{\pi}{2}} \frac{\cot \theta}{\frac{\pi}{2} - \theta} \]

we can follow these steps:

### Step 1: Substitute the limit
We start by substituting \(\theta = \frac{\pi}{2}\) into the expression:

\[ \cot\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \frac{\pi}{2} - \frac{\pi}{2} = 0 \]

This gives us the indeterminate form \(\frac{0}{0}\).

### Step 2: Apply L'Hôpital's Rule
Since we have the indeterminate form \(\frac{0}{0}\), we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the denominator separately.

### Step 3: Differentiate the numerator and denominator
The derivative of the numerator \(\cot \theta\) is:

\[ \frac{d}{d\theta}(\cot \theta) = -\csc^2 \theta \]

The derivative of the denominator \(\frac{\pi}{2} - \theta\) is:

\[ \frac{d}{d\theta}\left(\frac{\pi}{2} - \theta\right) = -1 \]

### Step 4: Rewrite the limit
Now, we can rewrite the limit using the derivatives:

\[ \lim_{\theta \to \frac{\pi}{2}} \frac{-\csc^2 \theta}{-1} = \lim_{\theta \to \frac{\pi}{2}} \csc^2 \theta \]

### Step 5: Evaluate the limit
Now we need to evaluate \(\csc^2 \theta\) as \(\theta\) approaches \(\frac{\pi}{2}\):

\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \]

As \(\theta\) approaches \(\frac{\pi}{2}\), \(\sin \left(\frac{\pi}{2}\right) = 1\), so:

\[ \csc^2\left(\frac{\pi}{2}\right) = \frac{1}{1^2} = 1 \]

### Final Answer
Thus, the limit evaluates to:

\[ \lim_{\theta \to \frac{\pi}{2}} \frac{\cot \theta}{\frac{\pi}{2} - \theta} = 1 \]

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