Problem 43 - Contemporary Precalculus

Question

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow \pi / 2} x \cos x$$

Step-by-Step Solution

Answer

Answer: The limit of the function \(x\cos x\) as \(x\) approaches \(\pi/2\) is \(0\), and it is equal to the function's value at \(c = \pi/2\).

Step 1: Analyze the given function

We are given the function \(x \cos x\). Let's first examine the properties of the function to determine if it's continuous. The function is an algebraic combination of \(x\) and \(\cos x\). Both of these functions are continuous in their domains, so \(x \cos x\) is continuous everywhere as well.

Step 2: Evaluate the limit

As \(x\) approaches \(\pi/2\), we want to find the limit of the function. Since the function is continuous, we can directly substitute the value of \(x\) to get the limit: $$\lim _{x \rightarrow \pi / 2} x \cos x = \frac{\pi}{2} \cos \frac{\pi}{2}$$ Now, we know that \(\cos (\pi/2) = 0\). So, $$\lim _{x \rightarrow \pi / 2} x \cos x = \frac{\pi}{2} \cdot 0 = 0$$ So, the limit is \(0\).

Step 3: Compare the limit to the function's value

Let's find the value of the function at \(c = \pi/2\): $$f(c) = f(\pi/2) = \frac{\pi}{2} \cos \frac{\pi}{2} = \frac{\pi}{2} \cdot 0 = 0$$ Both the limit and the function's value are equal to \(0\). Therefore, $$\lim_{x\rightarrow \pi/2} x\cos x = f(\pi/2)$$ In conclusion, the limit of the function \(x\cos x\) as \(x\) approaches \(\pi/2\) is \(0\), and it is equal to \(f(\pi/2)\), the function's value at \(c = \pi/2\).

Key Concepts

Continuous Functions

In calculus, understanding whether a function is continuous or not is crucial for evaluating limits. A continuous function is one where small changes in the input produce small changes in the output. Think of a continuous function as a stretchy, unbroken curve on a graph.
For the function \(x \cos x\), both \(x\) and \(\cos x\) are individually continuous over their domains:

• The function \(x\) is a simple linear function, continuous everywhere on the real number line.
• The function \(\cos x\) is a trigonometric function, continuous at every point on its domain, which is all real numbers.

Since the product of continuous functions is continuous, \(x \cos x\) is continuous everywhere. This is why evaluating its limit at a specific point involves simply substituting that point into the function to find the result.
Continuity helps us confirm that the limit of the function as \(x\) approaches a specific value equals the function's actual value at that point.

Evaluating Limits

Evaluating limits in calculus is like predicting where a car will end up by knowing its path. When we know a function involves continuous behavior, finding the limit as \(x\) approaches a certain value becomes straightforward.
To evaluate \(\lim _{x \rightarrow \pi / 2} x \cos x\), given that \(x \cos x\) is continuous at \(x = \pi/2\), we substitute \(\pi/2\) directly into the function.

• This involves calculating \(\frac{\pi}{2} \cdot \cos(\frac{\pi}{2})\).
• Here, since \(\cos \frac{\pi}{2} = 0\), the product becomes \(0\); hence, the limit is \(0\).

The substitution method only works because the function is continuous at the point we're evaluating.
If the function weren't continuous, we would need additional techniques such as recognizing indeterminate forms or considering left-right limits to find the limit.

Trigonometric Functions

Trigonometric functions are fundamental to calculus, especially when dealing with periodic phenomena like waves. The cosine function, \(\cos x\), is a crucial trigonometric function that oscillates between \(-1\) and \(1\).
This function repeats every \(2\pi\) radians, displaying a continuous wavy pattern. In our limit problem, \(\cos x\) determines the behavior of \(x \cos x\) as \(x\) approaches \(\pi/2\).

• At \(x = \pi/2\), \(\cos(\pi/2)\) equals \(0\).
• This zero value plays a pivotal role, turning the product \(\frac{\pi}{2} \cdot 0\) into zero as well.

Understanding such properties of trigonometric functions, like when \(\cos x\) becomes zero, helps us determine limits involving these functions.
It reinforces why knowing the specific output of \(\cos x\) directly impacts the calculation of limits in expressions where it is a component.