Problem 51 - Calculus: Early Transcendentals

Question

Evaluate limit. $$\lim _{x \rightarrow \pi} \frac{\cos ^{2} x+3 \cos x+2}{\cos x+1}$$

Step-by-Step Solution

Answer

Function: $$\frac{\cos^2x + 3\cos x + 2}{\cos x + 1}$$ Answer: The limit of the given function as x approaches π is 1.

Step 1: Factor the numerator

To start, let's factor out the numerator part of the function: $$\cos^2x + 3\cos x + 2 = (\cos x + 1)(\cos x + 2)$$ Now, the function can be written as: $$\frac{(\cos x + 1)(\cos x + 2)}{\cos x + 1}$$

Step 2: Simplify the expression

We can now cancel out the common factors in the numerator and the denominator: $$\frac{(\cos x + 1)(\cos x + 2)}{\cos x + 1} = \cos x + 2$$

Step 3: Evaluate the limit

We now need to find the limit of this simplified expression as x approaches π: $$\lim_{x \rightarrow \pi} (\cos x + 2)$$ We can use the fact that the cosine of π is -1: $$\lim_{x \rightarrow \pi} (\cos x + 2) = (\cos \pi + 2) = (-1 + 2) = 1$$ So the limit of the given expression as x approaches π is 1.

Key Concepts

Limit of a Function

The concept of the limit of a function is fundamental in calculus. It helps us understand the behavior of a function as the input value approaches a specific point. For example, when we say \( \lim_{x \to c} f(x) = L \), we mean that as \( x \) gets closer to \( c \), the values of \( f(x) \) get closer to \( L \). This is crucial when direct substitution into a function is not possible, possibly due to undefined operations like division by zero. Instead, limits allow us to "bypass" these obstacles to discover the true behavior of a function at certain points.

Factoring

Factoring is a method used to break down complex expressions into simpler, multiplyable parts. This technique is particularly helpful when dealing with polynomial expressions. In our exercise, we have the expression \( \cos^2 x + 3 \cos x + 2 \). Factoring transforms this into \( ( \cos x + 1 )( \cos x + 2 ) \).

By factoring, we can simplify the calculation of limits, especially when the given expression contains forms like \( \frac{0}{0} \). This form is indeterminate, can be simplified, potentially enabling limit evaluation through other means, like substitution.

Trigonometric Limits

Trigonometric limits deal with functions involving trigonometric functions such as sine, cosine, and tangent. These limits are common in calculus because trigonometric functions often arise in many mathematical contexts.

In the given exercise, the cosine function plays a crucial role. To evaluate limits involving trigonometric functions, you may need to recall specific angle values. For instance, knowing that \( \cos(\pi) = -1 \) enables us to evaluate \( \lim_{x \rightarrow \pi} (\cos x + 2) \) by substitution, giving us the result \( 1 \). Many standard limits connect trigonometric functions to real-world applications.

Simplifying Expressions

Simplifying expressions is about reducing complexity while keeping equivalence. It's like cleaning up a messy calculation to reveal the underlying essence of the problem. The goal is to make mathematical expressions easier to understand and work with.

After factoring the original expression, we obtained \( (\cos x + 1)(\cos x + 2) / (\cos x + 1) \). By canceling the common \( \cos x + 1 \) term, we simplified the expression to \( \cos x + 2 \).

This simplification is key to evaluating the limit efficiently and correctly, resulting in us finding a solution without extensive work. Proper simplification can reveal straightforward paths to solutions, promoting clearer mathematical thinking and problem-solving.