Problem 6 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow-\pi / 3} 3 x^{2} \cos x $$

Step-by-Step Solution

Answer

The limit is \(\pi^2/6\).

Step 1: Substitute the limit point into the expression

To evaluate the limit, substitute the given value for which x approaches (\(x \rightarrow -\pi/3\)) into the expression. So, replace \(x\) with \(-\pi/3\) in the expression \(3x^2 \cos x\). This results in \(3(-\pi/3)^2\cos(-\pi/3)\).

Step 2: Simplify the square term

Calculate \(-\pi/3\) squared. This results in \( (-\pi/3)^2 = \pi^2/9\).

Step 3: Evaluate the cosine function

Determine the cosine of \(-\pi/3\). Since cosine is an even function, \(\cos(-x) = \cos(x)\). Thus, \(\cos(-\pi/3) = \cos(\pi/3) = 1/2\).

Step 4: Multiply the components together

Now substitute these values back into the expression obtained in Step 1: \(3(\pi^2/9)(1/2)\). This simplifies to \((\pi^2/6) \).

Key Concepts

Trigonometric Limits

Understanding trigonometric limits is crucial when you're dealing with functions that involve sine, cosine, tangent or other trigonometric terms as they approach infinity or a particular point. Here, we examine how cosine is central to the calculation. Trigonometric functions often have special attributes. For instance, cosine is classified as an 'even' function. This means that \(\cos(-x) = \cos(x) \).If we apply this to our problem, when \(x\) approaches \(-\pi/3\), the term \(\cos(-\pi/3)\) is calculated as \(\cos(\pi/3)\). This simplification is pivotal because it reflects the symmetry of the cosine function around the y-axis. Knowing the properties of trigonometric functions can significantly ease the limit evaluation process.Recognizing the unique behavior of trigonometric functions at various points can help predict outcomes and simplify expressions before you reach the final solution. This knowledge is essential as it can reduce complex algebraic simplifications.

Calculus Steps

When evaluating limits in calculus, it's all about breaking the problem into manageable parts. This is exactly what step-by-step approaches in calculus enable us to do.

• Substitution: Initially, you substitute the limit point into the expression. Here, that meant plugging \(-\pi/3\) into the expression \(3x^2 \cos x\).
• Computation of Trigonometric Values: Calculating trigonometric values simplifies tasks further. In this problem, recognizing that \(\cos(-\pi/3) = 1/2\) due to the cosine's even nature illustrates this well.
• Multiplication and Arrangement: Finally, regroup different parts and simplify. The solution involves a straightforward multiplication followed by simplification: \(3\cdot (\pi^2/9) \cdot 1/2\) translates to the neat result, \(\pi^2/6\).

Understanding each of these steps offers insight into the logic behind calculus problem-solving and prepares students to tackle more complex limits.

Limit Substitution

Limit substitution is a fundamental technique used in calculus to find the value a function approaches as the input approaches a specified point. It involves directly replacing the variable in the function with the number it approaches. For this exercise, the limit point is \(-\pi/3\). First, substitute \(x\) with \(-\pi/3\) in the expression \(3x^2 \cos x\). Substitution often leads to direct computation, which simplifies the process.

• Benefits: Substitution provides a quick route to an answer, especially when the function is continuous at that point. There's no need for complex manipulations or transformations.
• Challenges: This method may not always work straightforwardly for discontinuous functions or indeterminate forms, where further transformations or specialized techniques like L'Hôpital's Rule may be necessary.

In cases like this problem, limit substitution directly leads to the answer without needing special consideration of undefined points, showcasing its effectiveness when applicable.