Problem 35 - Calculus for Biology and Medicine

Question

Use a graphing calculator to investigate $$ \lim _{x \rightarrow 1} \sin \frac{1}{x-1} $$

Step-by-Step Solution

Answer

The limit \( \lim _{x \rightarrow 1} \sin \frac{1}{x-1} \) does not exist.

Step 1: Understanding the Limit

We need to find \( \lim_{x \rightarrow 1} \sin \frac{1}{x-1} \). This limit is evaluated as \( x \) approaches 1, meaning we observe the behavior of the expression \( \sin \frac{1}{x-1} \) when \( x \) gets very close to 1 from both sides.

Step 2: Understanding the Expression

The expression \( \frac{1}{x-1} \) becomes undefined when \( x = 1 \), as it would result in division by zero. As \( x \) gets closer to 1, \( \frac{1}{x-1} \) diverges to \( \pm \infty \). Thus, we need to investigate what happens to \( \sin \frac{1}{x-1} \) as it approaches infinite oscillations.

Step 3: Investigating with Graphing

Using a graphing calculator, graph the function \( y = \sin \frac{1}{x-1} \) around \( x=1 \), specifically examining the intervals on either side of \( x=1 \). As \( x \) approaches 1, \( \sin \frac{1}{x-1} \) oscillates between -1 and 1 very rapidly due to the high-frequency oscillations of the sine function as its argument goes to infinity.

Step 4: Conclusion from Graphical Investigation

The graph shows that there is no single value that \( \sin \frac{1}{x-1} \) approaches as \( x \rightarrow 1 \). Instead, \( \sin \frac{1}{x-1} \) continues to oscillate between -1 and 1. Hence, the limit does not exist because there is no convergence to a particular value.

Key Concepts

graphing calculator

A graphing calculator is a fantastic tool that helps in understanding complex functions and their behavior visually. When dealing with limits, especially those involving trigonometric functions like sine, graphing calculators are invaluable. They can quickly illustrate how functions behave as they approach certain points. In the exercise involving \[ \lim _{x \rightarrow 1} \sin \frac{1}{x-1} \] we use a graphing calculator to plot the function. This visual exploration helps us see patterns and oscillations that the function exhibits as \( x \) approaches 1. By observing the graph, you'll find that the sine function oscillates quickly between

• -1 and 1

as \( x \) nears 1, enhancing understanding of the underlying mathematical behavior. A graphing calculator makes these complex oscillations obvious, something numerical analysis might miss.

infinite oscillations

Trigonometric functions like sine are known for their oscillatory nature. As \( x \) approaches 1 in our limit problem, \[ \sin \frac{1}{x-1} \]achieves very high frequency oscillations. The term \(\frac{1}{x-1}\)increases to positive or negative infinity, causing the sine function's input to vary wildly in a very short interval. This leads to what we call infinite oscillations. Infinite oscillations occur when a function swings back and forth

• infinitely fast
• without settling down

to a fixed value. In sine’s case, the rapid oscillation means it never stabilizes to a single number, hence for the limit, a single value doesn't exist. Understanding how this impacts the limit helps clarify why the function behaves unpredictably near \( x \)=1.

graphical investigation

Graphical investigation provides a visual method to analyze and understand limits and their behaviors. Instead of relying solely on algebra, plotting a graph uncovers nuances that numbers alone might obscure. For the given limit \[ \lim _{x \rightarrow 1} \sin \frac{1}{x-1} \]we use a graph to visually showcase the behavior of the function near \(x=1\). Here’s what a graphical investigation involving graphing calculators typically involves:

• Plotting the function over a domain close to the critical point, \(x=1\).
• Observing patterns, such as infinite oscillations.
• Determining whether the function seems to approach a single value or not.

In our exercise, the graph demonstrates that \(\sin \frac{1}{x-1}\) fluctuates between -1 and 1 indefinitely without approaching a specific limit. Such graphical observations confirm that the limit doesn't exist due to the continuous oscillation. This type of investigation greatly aids students in visually comprehending complex behaviors of functions.