Problem 29 - Calculus: Early Transcendentals

Question

a. \(\lim _{x \rightarrow 2^{+}} \frac{1}{\sqrt{x(x-2)}}\) b. \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x(x-2)}}\) c. \(\lim _{x \rightarrow 2} \frac{1}{\sqrt{x(x-2)}}\)

Step-by-Step Solution

Answer

Answer: The limit is undefined as x approaches 2 from both the right and the left, and therefore, the overall limit as x approaches 2 is also undefined.

Step 1: Analyze the function when x approaches 2 from the right

When x approaches 2 from the right (e.g. 2.001, 2.0001, etc.), we can see that \(x\) is always greater than 2. Thus, \((x-2)\) will be positive but very close to 0.

Step 2: Calculate the function when x approaches 2 from the right

As we have the function \(\frac{1}{\sqrt{x(x-2)}}\), when x approaches 2 from the right, we have the value of the function approaching \(\frac{1}{\sqrt{2(2-2)}}\) which simplifies to \(\frac{1}{\sqrt{0}}\). However, this situation creates a dilemma as we cannot divide by 0. Therefore, the limit as x approaches 2 from the right is undefined. b. \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x(x-2)}}\)

Step 1: Analyze the function when x approaches 2 from the left

When x approaches 2 from the left (e.g. 1.999, 1.9999, etc.), we can see that \(x\) is always less than 2. Thus, \((x-2)\) will be negative.

Step 2: Calculate the function when x approaches 2 from the left

As we have the function \(\frac{1}{\sqrt{x(x-2)}}\), when x approaches 2 from the left, the term \((x-2)\) is negative, and since a negative value is an input to a square root, the function becomes undefined. Therefore, the limit as x approaches 2 from the left is undefined. c. \(\lim _{x \rightarrow 2} \frac{1}{\sqrt{x(x-2)}}\)

Step 1: Analyze the function when x approaches 2

In the previous two parts (a and b), we found that the limit of the function when x approaches 2 from the right is undefined and the limit when x approaches 2 from the left is undefined as well.

Step 2: Determine the limit when x approaches 2

Since both left and right limits are undefined, the two-sided limit does not exist. Therefore, the limit as x approaches 2 is undefined.

Key Concepts

One-Sided Limits

One-sided limits are a key concept in understanding how functions behave as they approach a specific point from one direction. This is particularly useful when analyzing functions that have different behaviors on either side of a point. A one-sided limit focuses on the direction from which a variable approaches a value.

• When the input variable approaches a constant from the right (e.g., numbers slightly larger than the constant like 2.001), we call it a "right-hand limit." This is often denoted as \(\lim_{x \to a^{+}} f(x)\).
• If the variable approaches from the left (e.g., numbers slightly smaller than the constant like 1.999), it is termed a "left-hand limit," represented as \(\lim_{x \to a^{-}} f(x)\).

One-sided limits help us understand functions that are not continuous at a point, or where the behavior differs significantly from either side. When both these one-sided limits don't result in the same value, it means the two-sided limit does not exist. This is a crucial aspect to consider in calculus.

Undefined Limits

The term "undefined limit" is used when a function's limit does not approach a specific number as the input approaches a particular value. This can happen due to several reasons:

• When factoring results in division by zero.
• When evaluating a square root of a negative number, leading to imaginary results.
• When the function grows with no bound, therefore not approaching any finite limit.

In this exercise, as we approach 2, both from the left and the right, we encounter undefined situations.

• Approaching from the right results in a division by zero, since \((x-2)\) becomes 0.
• From the left, the expression inside the square root becomes negative, which makes the function undefined.

Understanding when and why a limit is undefined is fundamental, as it signals areas where the function may have discontinuities or special behavior that require further exploration.

Square Root Function

The square root function, denoted as \(\sqrt{x}\), takes a non-negative number and provides its square root. This function has specific rules:

• It is only defined for non-negative numbers, as square roots of negative numbers are not real.
• The output of \(\sqrt{x}\) is always non-negative.

In the problem we are analyzing, the square root function is part of a complex expression. This highlights special considerations, particularly as input values approach the point where the square inside the root may turn negative, thus rendering the function undefined. When working with square root functions, always consider the domain to ensure the input values do not violate the function's non-negativity requirement, particularly when they form part of larger expressions.

Approaching Limits from Left and Right

When evaluating limits, approaching from both the left and the right helps us understand the complete behavior of the function around a point. This approach is essential to determine if a function is continuous or if there is a break or asymptote at that point.

• Approaching from the right (right-hand limit) involves considering values slightly greater than the point of interest.
• Approaching from the left (left-hand limit) involves values slightly lesser.

This problem illustrates that when approaching the number 2 from both sides, unique situations occur:

• From the right, we seek values like 2.001 and note that division by zero arises.
• From the left, with values like 1.999, the square root takes a negative, which is undefined.

Understanding both approaches lets us see clearer that at x = 2, the function experiences conditions that make traditional limit evaluations not applicable, pointing to a breakdown in conventional continuity or behavior.