Evaluate the following :
`lim_(x to 0)xsecx`

Answer

To evaluate the limit \( \lim_{x \to 0} x \sec x \), we can follow these steps:

### Step 1: Rewrite the expression
The secant function is defined as:
\[ \sec x = \frac{1}{\cos x} \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 0} x \sec x = \lim_{x \to 0} x \cdot \frac{1}{\cos x} = \lim_{x \to 0} \frac{x}{\cos x} \]

### Step 2: Evaluate the limit
Now, we can substitute \( x = 0 \) into the expression:
\[ \lim_{x \to 0} \frac{x}{\cos x} = \frac{0}{\cos(0)} \]
Since \( \cos(0) = 1 \), we have:
\[ \frac{0}{1} = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} x \sec x = 0 \]