Chapter 2: Problem 29
a. \(\lim _{x \rightarrow 2^{+}} \frac{1}{\sqrt{x(x-2)}}\) b. \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x(x-2)}}\) c. \(\lim _{x \rightarrow 2} \frac{1}{\sqrt{x(x-2)}}\)
Short Answer
Expert verified
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 by step solution
01
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.
02
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. \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x(x-2)}}\)
03
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.
04
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. \(\lim _{x \rightarrow 2} \frac{1}{\sqrt{x(x-2)}}\)
05
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.
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \(\lim_{x \to a^{-}} f(x)\).
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.
- 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.
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.
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.
- 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.