`Lim_(x to pi) (1+sec^(3)x)/(tan^(2)x)`

Answer

To solve the limit \( \lim_{x \to \pi} \frac{1 + \sec^3 x}{\tan^2 x} \), we will follow these steps:

### Step 1: Substitute \( x = \pi \)
First, we substitute \( x = \pi \) into the expression to check the form of the limit.

\[ \sec(\pi) = -1 \quad \text{and} \quad \tan(\pi) = 0 \]

So, we have:

\[ 1 + \sec^3(\pi) = 1 + (-1)^3 = 1 - 1 = 0 \]

And,

\[ \tan^2(\pi) = 0^2 = 0 \]

Thus, the limit takes the form \( \frac{0}{0} \), which is indeterminate.

### Step 2: Factor the numerator
We will factor the numerator \( 1 + \sec^3 x \). We can use the identity for the sum of cubes:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

Here, let \( a = 1 \) and \( b = \sec x \):

\[ 1 + \sec^3 x = (1 + \sec x)(1 - \sec x + \sec^2 x) \]

### Step 3: Rewrite the limit
Now we can rewrite the limit:

\[ \lim_{x \to \pi} \frac{(1 + \sec x)(1 - \sec x + \sec^2 x)}{\tan^2 x} \]

### Step 4: Rewrite \( \tan^2 x \)
Recall that \( \tan^2 x = \sec^2 x - 1 \). Thus, we can rewrite the limit as:

\[ \lim_{x \to \pi} \frac{(1 + \sec x)(1 - \sec x + \sec^2 x)}{\sec^2 x - 1} \]

### Step 5: Simplify the expression
Notice that \( \sec^2 x - 1 = \tan^2 x \). So we can simplify further:

\[ \lim_{x \to \pi} \frac{(1 + \sec x)(1 - \sec x + \sec^2 x)}{\tan^2 x} \]

### Step 6: Cancel common factors
As \( x \to \pi \), both the numerator and denominator approach 0. Thus, we can cancel \( (1 + \sec x) \) from both the numerator and denominator:

\[ \lim_{x \to \pi} \frac{1 - \sec x + \sec^2 x}{\sec x - 1} \]

### Step 7: Substitute \( x = \pi \) again
Now substituting \( x = \pi \):

\[ 1 - \sec(\pi) + \sec^2(\pi) = 1 - (-1) + (-1)^2 = 1 + 1 + 1 = 3 \]

And,

\[ \sec(\pi) - 1 = -1 - 1 = -2 \]

### Step 8: Final limit calculation
Thus, we have:

\[ \lim_{x \to \pi} \frac{3}{-2} = -\frac{3}{2} \]

### Final Answer
The limit is:

\[ \boxed{-\frac{3}{2}} \]