Problem 14 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 10^{+}} \frac{|x-10|}{x-10} $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches \(10^{+}\) is \(1\).

Step 1: Understanding the Absolute Value Function

First, let's understand the absolute value function \(|x-10|\). When \(x\) is greater than or equal to \(10\), the expression \(|x-10|\) is equal to \(x-10\) because the result is nonnegative. When \(x\) is less than \(10\), \(|x-10|\) is equal to \(-(x-10)\) or \(10 - x\), this is because the result should be nonnegative.

Step 2: Substituting the Expression

Because we are finding the limit as \(x\) approaches \(10^{+}\) (from the right), \(x\) will always be greater than or equal to \(10\). Therefore, we can substitute \(|x-10|\) in our given expression with \(x-10\). The expression becomes: \[\lim _{x \rightarrow 10^{+}} \frac{x-10}{x-10}\]

Step 3: Simplifying the Expression

The expression \(\frac{x-10}{x-10}\) simplifies to \(1\). Therefore, the limit as \(x\) approaches \(10^{+}\) of the given function is \(1\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function, denoted \(|x-10|\), measures the distance of \(x\) from 10 on a number line. It strips away any sign, making the result non-negative.

Here’s how it works:

• If \(x \geq 10\), then \(|x-10| = x-10\). The outcome is positive or zero because the expression inside is positive or zero.
• If \(x < 10\), then \(|x-10| = -(x-10) = 10-x\). This makes it positive by flipping the sign.

In the given limit exercise, since we approach from the right (i.e., \(x \rightarrow 10^{+}\)), we're only dealing with \(x \geq 10\). Therefore, \(|x-10|\) simplifies directly to \(x-10\).

Exploring One-Sided Limits

One-sided limits help us understand the behavior of functions as they approach a certain point from only one direction. In our exercise, \(\lim_{x \rightarrow 10^{+}}\) directs us to consider values greater than or just approaching 10.

Why is this important?

• We only consider \(|x-10| = x-10\) in this scenario, simplifying our limit process.
• The limit as \(x\) approaches 10 from the right ensures we're analyzing behavior in this specific direction.

This helps focus our calculations by ignoring unnecessary cases, making it easier to determine the function's behavior and value at that approach.

Simplifying Expressions

Simplifying is crucial in calculus to make expressions more manageable. In this exercise, simplifying \(\frac{|x-10|}{x-10}\) when \(x \geq 10\) turns into \(\frac{x-10}{x-10}\).

This simplification leads to:

• \(\frac{x-10}{x-10} = 1\) when \(x - 10 eq 0\), meaning for all \(x eq 10\).
• The expression simplifies to 1 because the numerator and denominator are the same.

The limit is thus 1 as \(x\) approaches 10 from the right, making it easy to solve. Simplifying reduces complexity and reveals the function's true behavior at a particular point.