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Chapter 4: Problem 18

Find the vertical and horizontal asymptotes. $$f(x)=\sqrt{x}-\sqrt{x - 2}$$

Short Answer

Expert verified
The function \(f(x)=\sqrt{x}-\sqrt{x - 2}\) does not have any vertical asymptotes. The function has a horizontal asymptote at \(y = 0\) as \(x \rightarrow \infty\), considering the domain for \(x >= 2\).

Step by step solution

01

Finding Vertical Asymptotes

Vertical asymptotes occur when the denominator of a fraction equals zero, leading to undefined values. However, looking at the function \(f(x)=\sqrt{x}-\sqrt{x - 2}\), there's no fraction involved, thus, there's no vertical asymptote.
02

Finding Horizontal Asymptotes

Horizontal asymptotes are identified by studying the function as \(x \) tends to \(\pm \infty\). Both \(\sqrt{x}\) and \(\sqrt{x - 2}\) grow with x, but the latter lags by 2 units. As \(x\) increases, this difference becomes negligible, so the two terms cancel each other out, and \(f(x)\) approaches 0. Thus, the horizontal asymptote is \(y = 0\) as \(x\) goes to \(\pm \infty\).
03

Consider The Domain

However it is important to consider the domain of \(f(x)\), as \(x >= 2\). The expression \(\sqrt{x - 2}\) forces us to only consider \(x >= 2\). Therefore the correct limit is \(f(x) \rightarrow 0\) as \(x \rightarrow \infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
Vertical asymptotes are commonly associated with rational functions where the denominator equals zero, leading to undefined values. However, not all functions exhibit this behavior, as is the case with the function \( f(x) = \sqrt{x} - \sqrt{x - 2} \). There is no division by zero present as there are no fractions. This means there is no vertical asymptote in this function. It is important to identify the conditions that could cause vertical asymptotes in other scenarios:
  • If a function is rational, check for zeroes in the denominator.
  • If there is a square root, verify that the expression inside is positive to prevent undefined values.
While \( f(x) \) does not have a vertical asymptote, always think about whether there are other elements in similar functions that might create undefined points or restrictions to the graph.
Horizontal Asymptotes
Horizontal asymptotes inform us about the behavior of a function as \( x \) approaches infinity. For \( f(x) = \sqrt{x} - \sqrt{x-2} \), each term \( \sqrt{x} \) and \( \sqrt{x-2} \) grows towards infinity as \( x \) increases. However, \( \sqrt{x-2} \) has a slight delay compared to \( \sqrt{x} \) as it's pulled back by 2 units. Over large values of \( x \), this offset diminishes, causing the difference between the two terms to approach zero.
  • This results in a horizontal asymptote at \( y = 0 \) as \( x \) goes to infinity.
  • Remember: Horizontal asymptotes show us the potential settling line of the function.
  • Identify these by observing the leading behaviors of terms in the function as \( x \) extends to extreme values.
Understanding this will make you proficient in predicting long-term tendencies of functions.
Domain of a Function
The domain of a function refers to all possible input values (\( x \)) that allow the function to work without issues, such as undefined expressions. For the function \( f(x) = \sqrt{x} - \sqrt{x - 2} \), you must ensure the expressions under each square root remain non-negative.
  • \( \sqrt{x} \) is defined for \( x \geq 0 \).
  • \( \sqrt{x - 2} \) requires \( x - 2 \geq 0 \), resulting in \( x \geq 2 \).
Thus, the domain for this function is \( x \geq 2 \). This is crucial because it impacts where the function graph actually exists on a coordinate plane. Always verify domains to ensure that your function representations accurately reflect valid \( x \) values and potential limitations.

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