The value of `lim_(xto pi) (1+cos^(3)x)/(sin^(2)x)" is "`

Options

A. [2, 5)
B. (1, 5)
C. (-1, 5)
D. `(-oo, oo)`

Explanation

To solve the limit

\[ \lim_{x \to \pi} \frac{1 + \cos^3 x}{\sin^2 x}, \]

we can follow these steps:

### Step 1: Rewrite the numerator
We can express \(1 + \cos^3 x\) using the sum of cubes formula:

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

### Step 2: Rewrite the denominator
We know that \(\sin^2 x = 1 - \cos^2 x\). Therefore, we can rewrite the limit as:

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

### Step 3: Simplify the denominator
The denominator \(1 - \cos^2 x\) can be factored as:

\[ 1 - \cos^2 x = (1 - \cos x)(1 + \cos x). \]

### Step 4: Substitute and cancel terms
Now substituting this back into the limit gives us:

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

We can cancel \(1 + \cos x\) from the numerator and denominator (as long as \(x \neq \pi\)):

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

### Step 5: Evaluate the limit
Now, we can substitute \(x = \pi\):

- \(\cos \pi = -1\), so \(1 - \cos \pi = 1 - (-1) = 2\).
- \(\cos^2 \pi = (-1)^2 = 1\).

Substituting these values gives:

\[ \frac{1 - (-1) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}. \]

### Final Answer
Thus, the value of the limit is:

\[ \frac{3}{2}. \]

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