Problem 71 - Calculus Early Transcendentals

Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow \pi} \frac{\cos ^{2} x+3 \cos x+2}{\cos x+1}$$

Step-by-Step Solution

Answer

Answer: The limit of the given function as x approaches π is 1.

Step 1: Factorizing the numerator

Factorize the quadratic expression in the numerator: $$\cos^2 x + 3\cos x + 2 = (\cos x + 1)(\cos x + 2)$$

Step 2: Simplifying the given function

Substitute the factorized form of the numerator back into the given function: $$\frac{\cos^2 x + 3\cos x + 2}{\cos x + 1} = \frac{(\cos x + 1)(\cos x + 2)}{\cos x + 1}$$ Since the function is not in an indeterminate form 0/0 or ∞/∞, we can now cancel the common factor: $$\frac{(\cos x + 1)(\cos x + 2)}{\cos x + 1} = \cos x + 2$$

Step 3: Evaluate the limit

Now that the function is simplified, we can find the limit as x approaches π: $$\lim_{x \rightarrow \pi} (\cos x + 2) = \cos(\pi) + 2 = -1 + 2 = 1$$ The limit of the given function as x approaches π is 1.

Key Concepts

Limit Evaluation

In calculus, determining limits helps us understand the behavior of functions as they approach a certain point. Limit evaluation is crucial, especially when dealing with functions that aren't easily simplified or where direct substitution doesn't work. To evaluate a limit:

• Check if direct substitution gives a clear result. If it does, you have your solution.
• If substitution results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), look for ways to simplify the function.
• Apply algebraic techniques, such as factoring or using trigonometric identities, to simplify the expression.
• If necessary, apply limit laws or theorems (like L'Hôpital's Rule) for solving limits involving indeterminate forms.

In the context of the exercise, direct substitution might initially seem daunting due to the complex form of the function. However, factoring simplifies it, allowing for easy limit evaluation.

Factoring in Algebra

Factoring is an essential algebraic technique used to simplify expressions and solve equations, especially quadratics. With factoring, we restructure an expression into a product of simpler expressions or factors.
Factoring is especially useful in evaluating limits, as it helps simplify complex rational expressions. Here’s how we can approach factoring a quadratic expression, such as \(\cos^2 x + 3\cos x + 2\):

• Look for two numbers that multiply to give the constant term (2 here) and add up to the middle coefficient (3 in this case).
• Recognize patterns, such as a simple square (a + b)^2, or use them to trial combinations.
• Use the factored form to cancel common factors with the denominator, simplifying the overall expression for easier limit evaluation.

Once factored, limits can often be directly evaluated by substituting the variable's approaching value without concerns of division by zero or indeterminate forms.

Trigonometric Functions

Trigonometric functions, like \(\cos x\), \(\sin x\), and \(\tan x\), are fundamental in both algebra and calculus. They help describe relationships in triangles and periodic phenomena. When dealing with limits involving trigonometric functions, keep these tips in mind:

• Know the key trigonometric identities as they often help in simplifying expressions. For example, \(\cos^2 x + \sin^2 x = 1\) is a basic identity.
• Understand the behavior of these functions at key points, like 0, \(\pi/2\), and \(\pi\). These angles often provide neat results in limit evaluations.
• In the given exercise, recognizing that \(\cos \pi = -1\) was essential for finding the limit. Use trigonometric properties to simplify expressions for easy evaluation.
• Sometimes a limit involving trigonometric functions can be simplified to a constant or another known function, making the evaluation straightforward.

Mastering trigonometric functions in calculus aids not just in solving individual limit problems but also in understanding broader mathematical phenomena and multi-step problems in calculus.