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Chapter 2: Problem 6

Explain why the limits do not exist. $$\lim _{x \rightarrow 1} \frac{1}{x-1}$$

Short Answer

Expert verified
The limit does not exist because the function approaches \(+\infty\) from the right and \(-\infty\) from the left.

Step by step solution

01

Understand the concept of limits

When analyzing limits, we want to see where a function's value is heading as the input approaches a specific point. In this case, we are examining what happens to \( \frac{1}{x-1} \) as \( x \) approaches \( 1 \).
02

Analyze the behavior from the right

Consider the limit as \( x \) approaches 1 from the right (\( x \rightarrow 1^+ \)). When \( x \) is slightly greater than 1, \( x - 1 \) is positive but very small. Consequently, \( \frac{1}{x-1} \) becomes a large positive number because a small positive denominator results in a large positive value.
03

Analyze the behavior from the left

Now consider the limit as \( x \) approaches 1 from the left (\( x \rightarrow 1^- \)). When \( x \) is slightly less than 1, \( x - 1 \) is negative and very small. As a result, \( \frac{1}{x-1} \) becomes a large negative number because a small negative denominator results in a large negative value.
04

Conclusion about the limit

Since the values of the function approach \( +\infty \) as \( x \rightarrow 1^+ \) and approach \( -\infty \) as \( x \rightarrow 1^- \), the two one-sided limits do not match. Hence, the two-sided limit does not exist as \( x \rightarrow 1 \).

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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 crucial when determining the behavior of a function as it approaches a specific point from either side. For the function \( \frac{1}{x-1} \), as \( x \) approaches 1, we consider two separate scenarios: from the right (denoted as \( x \to 1^+ \)) and from the left (\( x \to 1^- \)). These one-sided limits help us understand the direction the function is heading as it gets close to the target value.
  • When approaching from the right (\( x \to 1^+ \)), \( x-1 \) is a tiny positive number, making \( \frac{1}{x-1} \) extremely large and positive.
  • Conversely, approaching from the left (\( x \to 1^- \)), \( x-1 \) turns into a small negative number, causing \( \frac{1}{x-1} \) to become significantly large and negative.
Understanding these one-sided limits highlights that when values on each side are different, the overall limit does not exist at that point.
Infinite limits
Infinite limits occur when a function's value grows unbounded as it approaches a certain point. In our example, the function \( \frac{1}{x-1} \) shows signs of infinite limits since its values move towards either positive infinity or negative infinity as \( x \) nears 1.
  • As \( x \to 1^+ \), \( \frac{1}{x-1} \) approaches positive infinity because the denominator becomes a very small positive number, causing the fraction to grow large without bound.
  • Conversely, as \( x \to 1^- \), \( \frac{1}{x-1} \) heads towards negative infinity due to the denominator being a very small negative value, making the fraction decrease endlessly.
These infinite behaviors indicate that at \( x=1 \), the values do not settle at a finite number but instead approach infinity, supporting why the limit does not exist.
Discontinuity
Discontinuity occurs in a function whenever there is a sudden break or an undefined point at a particular value. The function \( \frac{1}{x-1} \) experiences such a discontinuity at \( x=1 \) because the values from the different sides of the point do not converge to a single value.
  • The one-sided limits provide different values (positive and negative infinities), which do not equal one another.
  • Hence, there is no continuous path from one side of the point to the other. This abrupt change shows a discontinuity at \( x=1 \).
This kind of discontinuity, caused by an undefined point where the denominator is zero, is called a non-removable discontinuity, indicating that there is no way to "fix" this gap in the function.

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Most popular questions from this chapter

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