Determine the following limits.


a. lim x→1^+ x / |x − 1|

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the limit of the function \( \frac{x}{|x - 1|} \) as \( x \) approaches 1 from the right (denoted as \( x \to 1^+ \)).
2. Step 2: Analyze the behavior of the absolute value function. Since \( x \to 1^+ \), \( x \) is slightly greater than 1, making \( x - 1 \) positive. Therefore, \( |x - 1| = x - 1 \) in this region.
3. Step 3: Substitute the expression for the absolute value into the function. The function becomes \( \frac{x}{x - 1} \) for \( x > 1 \).
4. Step 4: Simplify the expression. As \( x \to 1^+ \), the denominator \( x - 1 \) approaches 0 from the positive side, while the numerator \( x \) approaches 1.
5. Step 5: Determine the behavior of the function. As \( x \to 1^+ \), \( \frac{x}{x - 1} \) tends to infinity, indicating that the limit is \( +\infty \).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as x approaches 1 from the right (denoted as x→1^+).

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In calculus, absolute values can affect the behavior of functions, especially at points where the expression inside the absolute value changes sign. For the limit in question, |x - 1| will behave differently depending on whether x is less than or greater than 1.

One-Sided Limits

One-sided limits are used to evaluate the behavior of a function as it approaches a specific point from one side only, either from the left (denoted as x→1^-) or from the right (denoted as x→1^+). This is crucial when dealing with functions that have different behaviors on either side of a point, such as discontinuities or vertical asymptotes. In this problem, we focus on the right-hand limit as x approaches 1.