Ascertain the existence of the following limits :
`lim_(x to pi//2) (tan x)/(sec x)`

Answer

To ascertain the existence of the limit \( \lim_{x \to \frac{\pi}{2}} \frac{\tan x}{\sec x} \), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the limit in terms of sine and cosine functions. Recall that:
\[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \sec x = \frac{1}{\cos x} \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to \frac{\pi}{2}} \frac{\tan x}{\sec x} = \lim_{x \to \frac{\pi}{2}} \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}} = \lim_{x \to \frac{\pi}{2}} \frac{\sin x}{\cos x} \cdot \cos x \]

### Step 2: Simplify the expression
Now, we can simplify the expression:
\[ \lim_{x \to \frac{\pi}{2}} \frac{\sin x}{\cos x} \cdot \cos x = \lim_{x \to \frac{\pi}{2}} \sin x \]

### Step 3: Evaluate the limit
Next, we evaluate the limit:
\[ \lim_{x \to \frac{\pi}{2}} \sin x \]
As \( x \) approaches \( \frac{\pi}{2} \), \( \sin x \) approaches \( 1 \).

### Conclusion
Thus, we conclude that:
\[ \lim_{x \to \frac{\pi}{2}} \frac{\tan x}{\sec x} = 1 \]
The limit exists and is equal to \( 1 \).

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