Problem 35 - Calculus for Biology and Medicine

Question

Use a graphing calculator to investigate $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Limit Expression

The given expression is \( \lim_{x \rightarrow 0} \frac{\sin x}{x} \). We want to find the limit of \( \frac{\sin x}{x} \) as \( x \) approaches 0.

Step 2: Set Up the Graph

To investigate this limit using a graph, enter the function \( y = \frac{\sin x}{x} \) into your graphing calculator. Ensure that the calculator is set to radian mode since trigonometric functions usually assume this.

Step 3: Observe the Behavior Near x=0

Graph the function and zoom in around \( x = 0 \). Observe the value of \( \frac{\sin x}{x} \) as \( x \) approaches 0 from both the positive and negative sides.

Step 4: Estimate the Limit from the Graph

Using the graph, note the behavior of the graph towards \( x = 0 \). You should see that as \( x \) approaches 0, the value of \( \frac{\sin x}{x} \) approaches 1.

Step 5: Conclusion from Graphical Investigation

Based on the graph, you can conclude that the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{x} \) is 1, as seen by the graph leveling out towards 1 as \( x \) gets closer to 0.

Key Concepts

Limit of a Function

Limits can be a tricky concept in calculus, but they are fundamental for understanding how functions behave at specific points. A limit essentially describes the value that a function approaches as the input approaches a certain point. For example, when we discuss the limit of a function as \( x \rightarrow c \), we are exploring what the function's output is getting closer to as \( x \) gets arbitrarily close to \( c \). To fully grasp this, remember:

• Approaching doesn't mean reaching: when evaluating limits, \( x \) approaches \( c \) but doesn't necessarily equal \( c \).
• One-sided limits: Sometimes, examining the behavior of a function from just the left or right can provide insight.
• If a function "levels off" or approaches a specific value, that value represents the limit at that point.

Understanding limits is key to delving into deeper aspects of calculus, such as derivatives and integrals, as it helps in modeling the change and areas under curves.

Trigonometric Limits

Trigonometric limits might seem complex at first, but they follow the same principles as regular limits, with some added intricacies due to trigonometric functions' periodic nature. One famous trigonometric limit to remember is:\[\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1\]This particular limit is foundational and is often used as a building block in calculus problems.
To understand why, consider these points:

• Sine and cosine functions oscillate between -1 and 1, making their limits unique.
• The small-angle approximation states that for small \( x \), \( \sin x \approx x \), which explains the result of the limit \( \frac{\sin x}{x} \).
• This concept is crucial in deriving many calculus results, including some differentiation rules.

Graphing Calculator Usage

Graphing calculators are powerful tools that can visualize complex mathematical concepts, making them more tangible. Here's how you can effectively use a graphing calculator to explore limits and functions:

• Inputting Functions: Start by entering the function you wish to analyze. For trigonometric limits, ensure your calculator is set to the correct mode (radians for functions involving \( \sin \) and \( \cos \)).
• Zoom & Trace: Utilize the zoom feature to closely inspect behavior, especially near points of interest like \( x = 0 \) in the case of trigonometric expressions. The trace function can provide exact values at particular points, helping in confirming limits.
• Observing Behavior: Graphs can show asymptotic behaviors, oscillations, or stability as \( x \rightarrow c \), aiding your intuitive understanding of limits.

By mastering these features, you can easily visualize results and verify your algebraic work, such as proving that \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \) through both graphical and analytical methods. This dual validation enhances comprehension and confidence in solving calculus challenges.