Evaluate each limit. 


$\underset{x\to \pi }{\lim }\frac{{\cos }^2\left(x\right)+3\cos \left(x\right)+2}{{\cos }^{}\left(x\right)+1}$

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to \pi} \frac{\cos^2(x) + 3\cos(x) + 2}{\cos(x) + 1} \).
2. Substitute \( x = \pi \) into the expression to check if it results in an indeterminate form.
3. Notice that \( \cos(\pi) = -1 \), so substitute \( \cos(x) = -1 \) into the numerator and denominator.
4. Simplify the numerator: \( \cos^2(x) + 3\cos(x) + 2 = (-1)^2 + 3(-1) + 2 = 1 - 3 + 2 = 0 \).
5. Simplify the denominator: \( \cos(x) + 1 = -1 + 1 = 0 \), indicating a \( \frac{0}{0} \) indeterminate form, so apply L'Hôpital's Rule or factor the expression to resolve the limit.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Trigonometric Functions

Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. In this limit problem, the cosine function is evaluated at the point x = π, which is essential for finding the limit. Understanding the properties and values of trigonometric functions at specific angles is key to solving such problems.

Indeterminate Forms

Indeterminate forms occur in calculus when direct substitution in a limit leads to expressions like 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation or L'Hôpital's rule, to resolve. Recognizing when a limit results in an indeterminate form is crucial for applying the appropriate techniques to evaluate it.