Chapter 2: Problem 14
In Exercises 13-20, use graphs and tables to find the limits. $$\lim _{x \rightarrow 2^{-}} \frac{x}{x-2}$$
Short Answer
Expert verified
\(\lim _{x \rightarrow 2^{-}} \frac{x}{x-2}\) is negative infinity.
Step by step solution
01
Determine the expression for the limit
The expression for this limit is given as \(\lim _{x \rightarrow 2^{-}} \frac{x}{x-2}\), where \(x\) is approaching 2 from the left.
02
Create a table to evaluate this limit
Prepare a table with values close to 2 from the left to understand the behavior of the function near \(x=2\). The table will look something like this: \n|x=|1.9|1.99|1.999|1.9999|\n|f(x)=|\|-19|-199|-1999|-19999|\nAs \(x\) approaches 2 from the left, the function \(f(x)\) appears to approach negative infinity.
03
Analyze the graph
If possible, plot the function \(f(x)=\frac{x}{x-2}\) and observe its behavior as \(x\) approaches 2 from the left. You can clearly observe that the function approaches negative infinity as \(x\) approaches 2 from the left, thus confirming our observation from the table.
04
Conclusion
According to both the table and graph analysis, as \(x\) approaches 2 from the left, \( \frac{x}{x-2} \) approaches negative infinity. Therefore, \(\lim _{x \rightarrow 2^{-}} \frac{x}{x-2}\) is negative infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-sided Limits
When calculating limits in calculus, it's crucial to distinguish between one-sided limits: the left-hand limit and the right-hand limit. A left-hand limit, denoted as \( \lim_{x \rightarrow c^-} f(x) \), refers to the value that a function \( f(x) \) approaches as \( x \) approaches \( c \) from the left. Similarly, a right-hand limit, denoted as \( \lim_{x \rightarrow c^+} f(x) \) refers to the behavior of \( f(x) \) as \( x \) approaches \( c \) from the right.
For the given exercise, we are interested in the left-hand limit as \( x \) approaches 2, which is represented by \( \lim_{x \rightarrow 2^-} \frac{x}{x-2} \). The negative superscript indicates that we are only considering values of \( x \) that are less than 2, getting infinitely close to it. When the function \( f(x) = \frac{x}{x-2} \) gets evaluated at numbers slightly less than 2, we notice that the values of \( f(x) \) decrease without bound, suggesting that the one-sided limit is \( -\infty \) as confirmed by our initial step by step solution.
To enhance understanding, one can visualize this by plotting points on a graph close to \( x=2 \) from the left and observing the trend as the \( x \) values get closer to 2. This graphical analysis aligns with our tabular findings where the function's value decreases sharply as we approach 2 from the left-hand side.
For the given exercise, we are interested in the left-hand limit as \( x \) approaches 2, which is represented by \( \lim_{x \rightarrow 2^-} \frac{x}{x-2} \). The negative superscript indicates that we are only considering values of \( x \) that are less than 2, getting infinitely close to it. When the function \( f(x) = \frac{x}{x-2} \) gets evaluated at numbers slightly less than 2, we notice that the values of \( f(x) \) decrease without bound, suggesting that the one-sided limit is \( -\infty \) as confirmed by our initial step by step solution.
To enhance understanding, one can visualize this by plotting points on a graph close to \( x=2 \) from the left and observing the trend as the \( x \) values get closer to 2. This graphical analysis aligns with our tabular findings where the function's value decreases sharply as we approach 2 from the left-hand side.
Asymptotic Behavior
The term 'asymptotic behavior' in mathematics refers to the way in which a graph of a function behaves as it gets very close to a certain line or point, but never actually reaches it. This line or point is commonly known as an 'asymptote'.
In the context of our exercise, \( \lim_{x \rightarrow 2^-} \frac{x}{x-2} \), the vertical line \( x=2 \) is a vertical asymptote of the function \( \frac{x}{x-2} \) because as \( x \) approaches 2 from the left, the function values become increasingly large in the negative direction. In other words, the function does not settle at a finite value but instead goes off towards \( -\infty \) indicating that the function is asymptotic to the vertical line \( x=2 \) as \( x \) goes to 2 from the left side.
To recognize this behavior, it can be useful to look at graphs or tables, as was done in Step 2 of our solution. Such observations allow us to conclude the nature of asymptotic behavior of the function around its asymptotes.
In the context of our exercise, \( \lim_{x \rightarrow 2^-} \frac{x}{x-2} \), the vertical line \( x=2 \) is a vertical asymptote of the function \( \frac{x}{x-2} \) because as \( x \) approaches 2 from the left, the function values become increasingly large in the negative direction. In other words, the function does not settle at a finite value but instead goes off towards \( -\infty \) indicating that the function is asymptotic to the vertical line \( x=2 \) as \( x \) goes to 2 from the left side.
To recognize this behavior, it can be useful to look at graphs or tables, as was done in Step 2 of our solution. Such observations allow us to conclude the nature of asymptotic behavior of the function around its asymptotes.
Graphical Analysis of Limits
Graphical analysis is a powerful tool for understanding the behavior of functions and their limits. By plotting the function on a graph, we can visually inspect the behavior as \( x \) approaches a particular value. This can be especially useful when discussing one-sided limits or asymptotic behavior, as these concepts can sometimes be counterintuitive.
In the given problem, plotting the function \( f(x) = \frac{x}{x-2} \) and zooming in on the point where \( x \) is approaching 2 from the left reveals that the graph plunges downwards towards negative infinity. This visual confirms our analytical finding from the table and clarifies the concept of a vertical asymptote.
The process outlined in Step 3 of our solution - creating a graph to observe the limit - is an example of how graphical analysis can confirm what we've deduced from tabular data. This is why graphing is often recommended in limit problems: it provides an intuitive and visual way of understanding the sometimes abstract concept of limits in calculus.
In the given problem, plotting the function \( f(x) = \frac{x}{x-2} \) and zooming in on the point where \( x \) is approaching 2 from the left reveals that the graph plunges downwards towards negative infinity. This visual confirms our analytical finding from the table and clarifies the concept of a vertical asymptote.
The process outlined in Step 3 of our solution - creating a graph to observe the limit - is an example of how graphical analysis can confirm what we've deduced from tabular data. This is why graphing is often recommended in limit problems: it provides an intuitive and visual way of understanding the sometimes abstract concept of limits in calculus.
