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

Answer

To evaluate the limit \( \lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta) \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting \( \sec \theta \) and \( \tan \theta \) in terms of sine and cosine:
\[ \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus, we can express the limit as:
\[ \sec \theta - \tan \theta = \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta} = \frac{1 - \sin \theta}{\cos \theta} \]

### Step 2: Substitute the limit
Now we substitute \( \theta \) approaching \( \frac{\pi}{2} \):
\[ \lim_{\theta \to \frac{\pi}{2}} \frac{1 - \sin \theta}{\cos \theta} \]

### Step 3: Evaluate the limit
As \( \theta \) approaches \( \frac{\pi}{2} \):
- \( \sin \theta \) approaches \( 1 \)
- \( \cos \theta \) approaches \( 0 \)

This gives us:
\[ \lim_{\theta \to \frac{\pi}{2}} \frac{1 - 1}{0} = \frac{0}{0} \]
This is an indeterminate form, so we can apply L'Hôpital's Rule.

### Step 4: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and the denominator:
- The derivative of the numerator \( 1 - \sin \theta \) is \( -\cos \theta \).
- The derivative of the denominator \( \cos \theta \) is \( -\sin \theta \).

Thus, we rewrite the limit:
\[ \lim_{\theta \to \frac{\pi}{2}} \frac{-\cos \theta}{-\sin \theta} = \lim_{\theta \to \frac{\pi}{2}} \frac{\cos \theta}{\sin \theta} \]

### Step 5: Evaluate the new limit
As \( \theta \) approaches \( \frac{\pi}{2} \):
- \( \cos \theta \) approaches \( 0 \)
- \( \sin \theta \) approaches \( 1 \)

So we have:
\[ \lim_{\theta \to \frac{\pi}{2}} \frac{\cos \theta}{\sin \theta} = \frac{0}{1} = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta) = 0 \]

---