`lim_(xrarr(pi//4))(sec^2x-2)/(tanx-1)` is

Options

A. 3
B. 1
C. 0
D. 2

Explanation

To solve the limit \( \lim_{x \to \frac{\pi}{4}} \frac{\sec^2 x - 2}{\tan x - 1} \), we will follow these steps:

### Step 1: Substitute \( x = \frac{\pi}{4} \)
First, we substitute \( x = \frac{\pi}{4} \) into the expression to check if we get an indeterminate form.

\[ \sec^2\left(\frac{\pi}{4}\right) = 2 \quad \text{and} \quad \tan\left(\frac{\pi}{4}\right) = 1 \]

Thus, we have:

\[ \sec^2\left(\frac{\pi}{4}\right) - 2 = 2 - 2 = 0 \]
\[ \tan\left(\frac{\pi}{4}\right) - 1 = 1 - 1 = 0 \]

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

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

where \( f(x) = \sec^2 x - 2 \) and \( g(x) = \tan x - 1 \).

### Step 3: Differentiate the Numerator and Denominator
Now we differentiate the numerator and denominator separately.

1. **Numerator**:
\[ f'(x) = \frac{d}{dx}(\sec^2 x - 2) = 2 \sec^2 x \tan x \]

2. **Denominator**:
\[ g'(x) = \frac{d}{dx}(\tan x - 1) = \sec^2 x \]

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

\[ \lim_{x \to \frac{\pi}{4}} \frac{2 \sec^2 x \tan x}{\sec^2 x} \]

### Step 5: Simplify the Expression
We can simplify this expression:

\[ \lim_{x \to \frac{\pi}{4}} 2 \tan x \]

### Step 6: Substitute Again
Now we substitute \( x = \frac{\pi}{4} \) into the simplified limit:

\[ 2 \tan\left(\frac{\pi}{4}\right) = 2 \cdot 1 = 2 \]

### Final Answer
Thus, the limit is:

\[ \boxed{2} \]