Problem 79 - Calculus: Early Transcendentals

Question

Horizontal and vertical asymptotes. a. Analyze \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\) analyze \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\sqrt{|x|}-\sqrt{|x-1|}$$

Step-by-Step Solution

Answer

Answer: There are no horizontal or vertical asymptotes for the given function \(f(x)=\sqrt{|x|}-\sqrt{|x-1|}\).

Step 1: Analyzing Horizontal Asymptotes

We'll start by analyzing the horizontal asymptotes, which can be done by finding the limit as \(x\) approaches positive and negative infinity. $$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} (\sqrt{|x|} - \sqrt{|x-1|})$$ $$\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} (\sqrt{|x|} - \sqrt{|x-1|})$$ For \(x \rightarrow \infty\): As \(x \rightarrow \infty\), \(|x| \rightarrow \infty\) and \(|x-1| \rightarrow \infty\). Therefore, both \(\sqrt{|x|}\) and \(\sqrt{|x-1|}\) are increasing without bound. For \(x \rightarrow -\infty\): As \(x \rightarrow -\infty\), \(|x| \rightarrow \infty\) and \(|x-1| \rightarrow \infty\). Therefore, both \(\sqrt{|x|}\) and \(\sqrt{|x-1|}\) are increasing without bound. Considering the limits for both positive and negative infinity, there are no horizontal asymptotes since both \(\sqrt{|x|}\) and \(\sqrt{|x-1|}\) are increasing without bound. Now we'll move on to analyzing the vertical asymptotes.

Step 2: Determining Points of Discontinuity

To determine the points of discontinuity that may result in vertical asymptotes, we'll consider the inside of each square root in the function. A vertical asymptote will occur when the denominator is equal to zero, and the numerator is different from zero. For \(f(x) = \sqrt{|x|} - \sqrt{|x-1|}\): 1. Analyze \(\sqrt{|x|}\): There is no point at which \(|x|\) would be zero. 2. Analyze \(\sqrt{|x-1|}\): The value inside the square root will be zero when \(x=1\). So, we might have a vertical asymptote at \(x=1\).

Step 3: Analyzing the Limit as x Approaches the Point of Discontinuity

Now we will analyze the limit as \(x\) approaches 1 from the left and from the right. $$\lim_{x \rightarrow 1^{-}} f(x) = \lim_{x \rightarrow 1^{-}} (\sqrt{|x|} - \sqrt{|x-1|})$$ $$\lim_{x \rightarrow 1^{+}} f(x) = \lim_{x \rightarrow 1^{+}} (\sqrt{|x|} - \sqrt{|x-1|})$$ As \(x\) approaches 1 from the left or from the right, both \(\sqrt{|x|}\) and \(\sqrt{|x-1|}\) will approach \(\sqrt{1} = 1\). Therefore, the difference between the two will approach 0. $$\lim_{x \rightarrow 1^{-}} f(x) = \lim_{x \rightarrow 1^{-}} (1 - 1) = 0$$ $$\lim_{x \rightarrow 1^{+}} f(x) = \lim_{x \rightarrow 1^{+}} (1 - 1) = 0$$ Since the limits as x approaches 1 from the left and from the right are equal to 0, there is no vertical asymptote at \(x=1\). In conclusion, there are no horizontal or vertical asymptotes for the given function \(f(x)=\sqrt{|x|}-\sqrt{|x-1|}\).

Key Concepts

horizontal asymptotes

When analyzing horizontal asymptotes, we are trying to find out the behavior of a function as it heads towards infinity or negative infinity. In simpler terms, horizontal asymptotes tell us where a graph is "flattening out" when we move far to the left or right on the coordinate plane.
For the function \(f(x) = \sqrt{|x|} - \sqrt{|x-1|}\), we looked at:

• \(\lim_{x \to \infty} f(x)\)
• \(\lim_{x \to -\infty} f(x)\)

In both cases, as \(x\) gets very large (either positively or negatively), both \(\sqrt{|x|}\) and \(\sqrt{|x-1|}\) increase without bound. This means the graph keeps climbing and doesn't level out. Thus, this function doesn't have any horizontal asymptotes. Horizontal asymptotes are very important in understanding how graphs behave on the extreme edges of the chart, but sometimes, just like here, a function might not have any.

vertical asymptotes

Vertical asymptotes occur where a function approaches infinity or negative infinity as the input gets close to a particular value. Finding vertical asymptotes helps us understand the behavior of a function near points of abrupt change or "holes."
For the function \(f(x) = \sqrt{|x|} - \sqrt{|x-1|}\), potential vertical asymptotes occur where the expressions inside the square roots become zero. Let's evaluate:

• \(\sqrt{|x|}\): \(|x|\) cannot be zero unless \(x = 0\), but this isn't an issue here.
• \(\sqrt{|x-1|}\): It's zero when \(x = 1\), indicating a potential asymptote.

However, when we check the limit as \(x\) approaches 1 from both sides (1 from the left and right), the results were finite (both equaled zero). Hence, there's no vertical asymptote at \(x=1\).
Vertical asymptotes often capture sudden spikes in the graph but require careful checking of limits to confirm them.

limits

In calculus, limits help us describe what a function does as it gets close to a specific point or reaches towards infinity. They're crucial for finding asymptotes and understanding function behavior.
For example, in the problem involving \(f(x) = \sqrt{|x|} - \sqrt{|x-1|}\):

• \(\lim_{x \rightarrow \infty} f(x)\) checks how the function behaves as \(x\) becomes very large.
• \(\lim_{x \rightarrow -\infty} f(x)\) does the same for very negative \(x\).
• \(\lim_{x \rightarrow 1^{-}} f(x)\) and \(\lim_{x \rightarrow 1^{+}} f(x)\) explore the function's behavior close to \(x = 1\).

Using limits, we found no horizontal or vertical asymptotes for this function. Limits provide the tools needed to approach tricky points and determine how a graph acts nearby, making them essential for exploring continuous and discontinuous behavior.

points of discontinuity

Points of discontinuity indicate where a function isn't smooth or continuous. These are spots where something unusual happens in the graph, such as a hole or a vertical asymptote.
In the function \(f(x) = \sqrt{|x|} - \sqrt{|x-1|}\), we investigated the potential discontinuity at \(x = 1\), where the expression \(\sqrt{|x-1|}\) becomes zero. However, distinct from a true discontinuity or asymptote, the limits from both sides of \(x = 1\) matched, meaning the function transitioned smoothly at that point without any breaks or jumps.
Understanding points of discontinuity is key to graphing functions accurately and predicting where a graph might have holes or abrupt changes. It helps us anticipate behavior that might not be initially obvious from the function's equation itself.