Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \sqrt{|x|} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Problem

We need to find the limit \( \lim_{x \rightarrow 0} \sqrt{|x|} \). This expression involves the square root of the absolute value of \( x \). The absolute value affects both the negative and positive paths towards zero.

Step 2: Consider Approaching from the Positive Side

First, consider the limit as \( x \to 0^+ \) (approaching zero from the positive side). The expression becomes \( \sqrt{x} \). As \( x \) approaches zero, the square root \( \sqrt{x} \) also approaches zero. Thus, \( \lim_{x \to 0^+} \sqrt{x} = 0 \).

Step 3: Consider Approaching from the Negative Side

Next, consider the limit as \( x \to 0^- \) (approaching zero from the negative side). Here \( |x| \) equals \(-x\), so the expression is \( \sqrt{-x} = \sqrt{|x|} \). As \( x \) approaches zero from the negative side, \( \sqrt{|x|} = \sqrt{-x} \) also approaches zero. Thus, \( \lim_{x \to 0^-} \sqrt{|x|} = 0 \).

Step 4: Finalizing the Conclusion

Since both one-sided limits (from 0^+ and 0^-) equal zero, the two-sided limit also equals zero. Therefore, \( \lim_{x \to 0} \sqrt{|x|} = 0 \).

Key Concepts

Absolute Value

In mathematics, the absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as \(|x|\). Absolute value is always non-negative. Here are some key points:

• If \(x\) is positive, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).
• If \(x\) is zero, \(|x| = 0\).

The absolute value just 'removes' any negative sign in front of a number, turning it into its positive counterpart. This simplifies expressions that may otherwise behave differently over different ranges of \(x\), such as in the calculation of limits. In our original exercise, the absolute value ensures the expression inside the square root is always non-negative, which allows us to take the square root directly for both negative and positive inputs, ultimately leading to both one-sided limits equating to zero.

Continuous Functions

A continuous function is one where small changes in input lead to small changes in output, with no abrupt jumps or breaks. In simple terms, you can draw the graph of a continuous function without lifting your pencil. This continuity ensures that limits can be effectively calculated.
When dealing with continuous functions, particularly when involving an absolute value, the function behaves predictably. In the exercise at hand, \(\sqrt{|x|}\) is continuous everywhere except at \(x = 0\). At this point, we look at the behavior from both sides (as \(x\) approaches 0 from the negative and positive sides). Since \(\sqrt{|x|}\) smoothly transitions through all values, it demonstrates continuity by yielding a consistent limit of 0 from both sides.

• Continuous functions simplify the process of finding limits as they behave predictably across their domain.
• Being able to rely on continuity across a function's domain allows for the simplification of calculations.

One-sided Limits

One-sided limits focus on investigating the behavior of a function as it approaches a specific point from one side only - either the left or the right.

• \(\lim_{x \to a^+} f(x)\) refers to the limit of function \(f(x)\) as \(x\) approaches \(a\) from the right (positive direction).
• \(\lim_{x \to a^-} f(x)\) refers to the limit of function \(f(x)\) as \(x\) approaches \(a\) from the left (negative direction).

This concept is particularly useful for functions that involve absolute values. In our exercise, we evaluated both one-sided limits:

• When approaching 0 from the positive side, \(x > 0\) and \(|x| = x\), so \(\lim_{x \to 0^+} \sqrt{x} = 0\).
• When approaching 0 from the negative side, \(x < 0\) and \(|x| = -x\), which ensures \(\lim_{x \to 0^-} \sqrt{|x|} = 0\).

Calculating one-sided limits helps in determining the behavior of functions around critical points, ensuring a thorough understanding of function behavior in calculus.