Problem 19
Question
GC In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{\sin x}{2 x} $$
Step-by-Step Solution
Verified Answer
The limit is \( \frac{1}{2} \).
1Step 1: Understanding the Limit Expression
We need to find the limit of \( \frac{\sin x}{2x} \) as \( x \) approaches 0. The expression involves \( \sin x \), which is a trigonometric function. At \( x = 0 \), the expression is in the form \( \frac{0}{0} \), which is indeterminate, suggesting the need for further analysis.
2Step 2: Apply L'Hôpital's Rule
L'Hôpital's Rule states that for limits resulting in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) forms, we can take the derivative of the numerator and the denominator separately. Here, the derivative of \( \sin x \) is \( \cos x \) and the derivative of \( 2x \) is \( 2 \). Therefore, the limit becomes: \[ \lim_{x \to 0} \frac{\cos x}{2} \] Since \( \cos 0 = 1 \), this simplifies to \( \frac{1}{2} \).
3Step 3: Confirm with a Graphing Calculator
To confirm the calculated limit, use a graphing calculator. Plot the function \( y = \frac{\sin x}{2x} \) and observe the behavior as \( x \) approaches 0. You should see that as you get closer to \( x = 0 \), the y-values approach \( \frac{1}{2} \), confirming our calculated limit.
Key Concepts
LimitsL'Hôpital's RuleTrigonometric Functions
Limits
Understanding the concept of limits is crucial in calculus. A limit helps in finding the value a function approaches as the input (in this case, \( x \)) gets closer to a certain point, such as 0. When we say \( \lim_{x \rightarrow 0} \frac{\sin x}{2 x} \), we want to see what value \( \frac{\sin x}{2x} \) nears as \( x \) approaches 0.
The main challenge with limits often arises when the function results in an indeterminate form, such as \( \frac{0}{0} \) as seen here. This means the expression doesn’t inherently reveal what value it’s approaching.
The main challenge with limits often arises when the function results in an indeterminate form, such as \( \frac{0}{0} \) as seen here. This means the expression doesn’t inherently reveal what value it’s approaching.
- To resolve this, methods like L'Hôpital's Rule or algebraic manipulation become very helpful.
- Limits can be easily estimated using a graphing calculator to visualize the function’s behavior near the limit point. This graphical method confirms the analytical solutions and ensures understanding from different perspectives.
L'Hôpital's Rule
L'Hôpital's Rule is a powerful tool for calculating limits that take an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The fundamental idea is to simplify the computation of a tricky limit by focusing on the derivatives of the numerator and denominator.
For the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{2x} \), applying L'Hôpital’s Rule means we find the derivative of \( \sin x \), which is \( \cos x \), and the derivative of \( 2x \), which is 2. This transforms our problem into a simpler one: \( \lim_{x \to 0} \frac{\cos x}{2} \).
For the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{2x} \), applying L'Hôpital’s Rule means we find the derivative of \( \sin x \), which is \( \cos x \), and the derivative of \( 2x \), which is 2. This transforms our problem into a simpler one: \( \lim_{x \to 0} \frac{\cos x}{2} \).
- With \( \cos 0 = 1 \), this simplifies directly to \( \frac{1}{2} \).
- L'Hôpital's Rule not only simplifies complex limits but also provides a systematic approach to evaluate them.
Trigonometric Functions
Trigonometric functions like \( \sin x \) and \( \cos x \) are fundamental in calculus due to their continuous and periodic nature. These functions repeatedly appear in problems involving limits, derivatives, and integrals.
In the context of the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{2 x} \), understanding the behavior of \( \sin x \) is essential.
In the context of the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{2 x} \), understanding the behavior of \( \sin x \) is essential.
- As \( x \) approaches 0, \( \sin x \) behaves very similarly to \( x \). In fact, \( \sin x \approx x \) for small values of \( x \), which is an underlying reason why L'Hôpital's Rule applies so effectively.
- The derivative of \( \sin x \) is \( \cos x \), which is part of the essential toolkit for solving advanced calculus problems.
Other exercises in this chapter
Problem 19
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