Problem 30 - Calculus: Early Transcendentals

Question

Estimating limits graphically and numerically Use a graph f \(f\) to estimate \(\lim f(x)\) or to show that the limit does not exist. Evaluate \(f(x)\) near \(x=a\) to support your conjecture. $$f(x)=\frac{3 \sin x-2 \cos x+2}{x} ; a=0$$

Step-by-Step Solution

Answer

Answer: The limit does not exist.

Step 1: Mapping the function

First, let's rewrite the given function, f(x), for better understanding. $$f(x) = \frac{3 \sin x - 2 \cos x + 2}{x}$$ Step 2: Evaluating the function values at different points near a

Step 2: Function values near a

We need to evaluate the function values for x close to 0 (both negative and positive) to see if the function approaches a unique value or not. Let's calculate \(f(x)\) for x = -0.1, -0.01, 0.01, and 0.1. For x = -0.1, \(f(-0.1) \approx \frac{3 \sin(-0.1) - 2 \cos(-0.1) + 2}{-0.1} \approx -7.133\) For x = -0.01, \(f(-0.01) \approx \frac{3 \sin(-0.01) - 2 \cos(-0.01) + 2}{-0.01} \approx -499.983\) For x = 0.01, \(f(0.01) \approx \frac{3 \sin(0.01) - 2 \cos(0.01) + 2}{0.01} \approx 500.017\) For x = 0.1, \(f(0.1) \approx \frac{3 \sin(0.1) - 2 \cos(0.1) + 2}{0.1} \approx 7.147\) Step 3: Graphing the function

Step 3: Graph analysis

Now, let's analyze the graph of the given function. After plotting the graph, we will find that there is a vertical asymptote at x = 0, which implies that the function does not approach any specific value as x approaches 0. Step 4: Drawing a conclusion

Step 4: Conclusion

Based on our numerical calculations and the graph analysis, we find that the function does not approach a unique value as x approaches 0. As a result, we can conclude that the limit does not exist: $$\lim_{x\to0} f(x) = \lim_{x\to0} \frac{3 \sin x - 2 \cos x + 2}{x} = \text{Does not exist}$$

Key Concepts

Graphical Limit Estimation

Graphical limit estimation involves analyzing the behavior of a function's graph as it approaches a specific value of x. By observing how the graph behaves near this point, we can guess whether a function converges to a particular number, diverges, or oscillates. In the context of the problem, plotting the graph of the function \( f(x) = \frac{3 \sin x - 2 \cos x + 2}{x} \) near \( x = 0 \) is crucial.

• Graph the function to visualize its behavior.
• Check for any breaks, sharp turns, or infinities at \( x = 0 \).
• Observe the function's behavior from both sides of the point \( x = 0 \).

When graphed, this function shows a vertical asymptote at \( x = 0 \). This graphical feature indicates that the limit \( \lim_{x \to 0} f(x) \) does not exist since the function does not settle at a specific value.

Numerical Limit Calculation

Numerical limit calculation involves evaluating the function values close to the point of interest, in this case \( a = 0 \), to see if these values approach a common number.
We calculate \( f(x) \) at values like \( x = -0.1, -0.01, 0.01, \) and \( 0.1 \). These calculations help us observe the behavior of the function numerically.

• \( f(-0.1) \approx -7.133 \)
• \( f(-0.01) \approx -499.983 \)
• \( f(0.01) \approx 500.017 \)
• \( f(0.1) \approx 7.147 \)

The rapid change in values as \( x \) approaches 0 from both sides supports the conclusion that the limit does not exist. The function doesn't stabilize towards a specific value, showing numerical divergence.

Functions with Asymptotes

Functions that have asymptotes often show limits that do not exist, especially vertical asymptotes. An asymptote is a line that the graph of a function approaches but never actually touches. In the given function, there is a vertical asymptote at \( x = 0 \). This is evident by observing both the graph and the numerical values.
Understanding asymptotes is crucial:

• Vertical asymptotes suggest that the function's values shoot off to infinity or negative infinity.
• They represent a limitation in the function's domain where a limit cannot exist due to unbounded behavior.

Recognizing this helps conclude why the limit \( \lim_{x \to 0} f(x) \) does not exist for this particular function.

Limit Existence

The existence of a limit depends on the function approaching a single, finite value as \( x \) approaches a specific point. In our exercise, the aim is to determine the behavior of the function \( f(x) = \frac{3 \sin x - 2 \cos x + 2}{x} \) as \( x \to 0 \). For a limit to exist, the left-hand limit \( \lim_{x \to 0^-} \) and the right-hand limit \( \lim_{x \to 0^+} \) must converge to the same value.
Steps to determine limit existence:

• Evaluate the behavior of \( f(x) \) from both negative and positive sides of \( a \).
• Check if the function values stabilize to a single number.

In this scenario, as seen from both graphical and numerical observations, the function diverges (values tend towards infinity), indicating no convergence to a single value. Therefore, the limit \( \lim_{x \to 0} f(x) \) does not exist.