Problem 47 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{x \rightarrow 2} \sin \left(\frac{1}{x}-\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

We have been given the expression \( \lim_{x \to 2} \sin \left( \frac{1}{x} - \frac{1}{2} \right) \) and need to find its limit as \( x \) approaches 2.

Step 2: Evaluate the Expression Inside Sine

First, substitute \( x = 2 \) in \( \frac{1}{x} - \frac{1}{2} \). This gives us \( \frac{1}{2} - \frac{1}{2} = 0 \).

Step 3: Apply the Limit to Sine

The limit simplifies to \( \lim_{x \to 2} \sin(0) \). Since \( \sin(0) = 0 \), this limit evaluates to 0.

Key Concepts

sine function

The sine function is a key concept in trigonometry, often abbreviated as "sin". It is a periodic function that oscillates between -1 and 1. The sine function is essential in various fields, such as physics and engineering, for modeling wave patterns and circular motion.

The sine function is defined as follows: for an angle \(\theta\) in a right triangle, it is the ratio of the length of the opposite side to the hypotenuse. When extended to the unit circle, it represents the y-coordinate of a point defined by an arc's length on the circle. The sine graph is a smooth, continuous wave crossing zero at intervals of \(n\pi\), where \(n\) is an integer. This property helps in evaluating limits involving the sine function.

In our problem, we focus on finding the limit of a sine function. Knowing that \( \sin(0) = 0 \) simplifies the limit evaluation significantly. When evaluating limits, this basic understanding of sine helps us ascertain the value quickly.

limit laws

Limit laws are a collection of rules that simplify the process of finding limits. They help manage complex expressions by breaking them into manageable parts. Understanding these laws is key to tackling a broad array of limit problems.

Here are some essential limit laws:

• The Constant Law: The limit of a constant is the constant itself. \(\lim_{x \to c} a = a\).
• The Sum/Difference Law: The limit of a sum or difference can be found by taking the sum or difference of the limits. \(\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)\).
• The Product Law: The limit of a product is the product of the limits. \(\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)\).
• The Quotient Law: The limit of a quotient \(\frac{f(x)}{g(x)}\) is \(\frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}\) given that \(\lim_{x \to c} g(x) eq 0\).

In our problem, we primarily used the substitution property and the fact that \(\sin(0) = 0\), thus applying these limit laws allows us to evaluate the limit effectively. Grasping these laws is fundamental to mastering calculus.

trigonometric limits

Trigonometric limits involve finding the limit of expressions that contain trigonometric functions like sine, cosine, or tangent, as a variable approaches a specific point. Often, these problems require a good understanding of trigonometric identities and limits.

Some common trigonometric limits include:

• \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
• \(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\)

When approaching trigonometric limits, it is often helpful to simplify the expression using identities or substitutions, just like we did by evaluating \(\lim_{x \to 2} \sin\left(\frac{1}{x} - \frac{1}{2}\right)\). By substituting \(x = 2\) early in the process, we reduced the expression inside the sine function to 0, allowing us to immediately know the result.Trigonometric limits, thus, push us to recognize patterns and apply known results, leading to quicker and more accurate evaluations.