Problem 50 - College Algebra and Calculus: An Applied Approach

Question

find the limit $$ \lim _{x \rightarrow 2} \frac{|x-2|}{x-2} $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches 2 does not exist, because the limit from the left is \(-1\) and from the right is \(1\).

Step 1: Break Down of The Absolute Value

The expression inside the absolute value, \(x-2\), changes sign at \(x = 2\). We often break down absolute values into a piecewise function. So \(|x-2|\) can be defined as: \(-(x-2)\), for \(x<2\) and \((x-2)\), for \(x>2\).

Step 2: Find the Limit From the Left

To find the limit as \(x\) approaches 2 from the left (\(x<2\)), we substitute \(-(x-2)\) into the absolute value part of the expression giving \(\frac{-(x-2)}{x-2} = -1\). Since \(-1\) is a constant, the limit as \(x\) approaches 2 from the left is \(-1\).

Step 3: Find the Limit From the Right

To find the limit as \(x\) approaches 2 from the right (\(x>2\)), we substitute \((x-2)\) into the absolute value part of the expression giving \(\frac{(x-2)}{x-2} = 1\). Hence the limit as \(x\) approaches 2 from the right is \(1\).

Step 4: Compare the Limits

Since the limit from the left is \(-1\) and the limit from the right is \(1\), and they are different, the limit of the function as \(x\) approaches 2 does not exist.

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. For example, the absolute value of both \(3\) and \( -3\) is \(3\) because both points are three units away from zero. Mathematically, absolute value is defined as:
\[ |x| = \left\{ \begin{array}{ll} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{array} \right. \]
In calculus, the concept of absolute value can lead to piecewise functions when determining limits, as we often consider the behavior of the function from both the left and the right side of a point.

Limits and Continuity

The concept of limits is fundamental in calculus and refers to the value that a function \(f(x)\) approaches as the input \(x\) approaches a specified value. Continuity, on the other hand, implies that a function is unbroken or uninterrupted over an interval.

A function is continuous at a point \(x = a\) if the following conditions are met:

• The function \(f(a)\) is defined.
• The limit of \(f(x)\) as \(x\) approaches \(a\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(f(a)\).

In the case of our exercise, the function is not continuous at \(x = 2\) because the limits from the left and the right do not match, which means the limit does not exist at that point.

Piecewise Functions

Piecewise functions are defined by different expressions based on different intervals of the input variable. They are particularly useful when a function behaves differently across various sections of its domain. By breaking the function into 'pieces', we can more easily analyze and understand its behavior.

Example in Calculus

In the context of our problem, the absolute value function \( |x-2| \) is redefined as a piecewise function depending on whether \(x < 2\) or \(x > 2\) to evaluate the limit at \(x = 2\). Thus piecewise functions contribute significantly to the understanding of limits for functions with absolute values.

Calculus

Calculus is the mathematical study of continuous change and covers a vast range of functions and behaviors. It is divided into two major branches: differential calculus and integral calculus.

Differential calculus deals with the concept of a derivative, which gives the rate at which a quantity changes, while integral calculus focuses on the concept of an integral, which measures the accumulation of quantities.

• In the domain of differential calculus, limits are used to define derivatives.
• In integral calculus, limits are used to find the area under curves.

Understanding the limit of a piecewise function, as demonstrated in the exercise, is a part of differential calculus, which is used to analyze the behavior of functions at specific points within their domain.