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Chapter 2: Problem 6

In Exercises 5 and \(6,\) explain why the limits do not exist. $$\lim _{x \rightarrow 1} \frac{1}{x-1}$$

Short Answer

Expert verified
The limits from the left and right are not equal; hence, the limit does not exist.

Step by step solution

01

Understand the Limit

We are asked to find the limit \( \lim _{x \rightarrow 1} \frac{1}{x-1} \). This involves evaluating the behavior of the function \( \frac{1}{x-1} \) as \( x \) approaches 1.
02

Analyze Approach from the Left Side

Consider approaching \( x = 1 \) from the left, i.e., values of \( x \) close to 1 but less than 1. For such \( x \), \( x-1 \) is negative, thus \( \frac{1}{x-1} \) becomes a large negative number. As \( x \rightarrow 1^- \), \( \frac{1}{x-1} \rightarrow -\infty \).
03

Analyze Approach from the Right Side

Now consider approaching \( x = 1 \) from the right, i.e., values of \( x \) greater than 1. For such \( x \), \( x-1 \) is positive, and \( \frac{1}{x-1} \) becomes a large positive number. As \( x \rightarrow 1^+ \), \( \frac{1}{x-1} \rightarrow \infty \).
04

Conclude the Non-existence of the Limit

Notice that approaching 1 from the left (\( x \rightarrow 1^- \)) results in \( -\infty \) while approaching from the right (\( x \rightarrow 1^+ \)) results in \( \infty \). Because the left-hand limit and the right-hand limit are not equal, the limit \( \lim _{x \rightarrow 1} \frac{1}{x-1} \) does not exist.

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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 explore what happens to a function as the variable approaches a specific value from one direction.
In limit problems, this means evaluating the behavior of the function as you approach the target value from either the left or the right.
  • **Left-Hand Limit:** Denoted as \(x \to a^-\), this examines what happens as \(x\) approaches \(a\) from values less than \(a\). If \(f(x)\) approaches a finite number, then the left-hand limit exists and equals that number.
  • **Right-Hand Limit:** Denoted as \(x \to a^+\), this looks at what happens as \(x\) approaches \(a\) from values greater than \(a\). If \(f(x)\) approaches a finite number, then the right-hand limit exists and equals that number.
For the function \(\frac{1}{x-1}\), analyzing the one-sided limits at \(x = 1\) reveals different behaviors.
Approaching from the left, \(x \to 1^-\), the function takes on increasingly negative values, heading toward \(-\infty\).
From the right, \(x \to 1^+\), the function becomes positive and approaches \(\infty\).
These differing results from each side contribute to a limit that does not agree.
Infinite Limits
Infinite limits occur when the values of a function grow indefinitely large as the variable approaches a particular point.
Understanding this behavior is crucial to analyzing how functions behave at points that might lead to astronomical values.
  • **Positive Infinity:** This happens when as \(x\) approaches a particular point, \(f(x)\) increases without bound, denoted as \(\lim_{x \to a} f(x) = \infty\).
  • **Negative Infinity:** Conversely, if \(f(x)\) decreases without bound as \(x\) approaches a point, it is described as \(\lim_{x \to a} f(x) = -\infty\).
The function \(\frac{1}{x-1}\) illustrates both infinite behaviors as \(x\) approaches 1.
Approaching from the left, it exhibits a negative infinite limit, \(-\infty\), and from the right, a positive infinite limit, \(\infty\).
These infinite limits help explain why the overall limit does not exist.
Limit Does Not Exist
In calculus, sometimes calculating the limit results in a situation where it simply does not exist.
This typically happens when one or more of the following occur:
  • **Disparity Between One-Sided Limits:** If the left-hand and right-hand limits are unequal, as in \(\lim_{x \to 1^-} \frac{1}{x-1} = -\infty\) and \(\lim_{x \to 1^+} \frac{1}{x-1} = \infty\), the limit doesn't exist.
  • **Unbounded Growth:** If the function approaches infinity (positive or negative) as it nears a specific \(x\)-value, like in our example, the limit can be said to not exist.
  • **Oscillation:** Sometimes, functions oscillate wildly within a limited interval and thus evade settling into any single value.
In our given function example, \(\frac{1}{x-1}\), the one-sided limits both approach infinity, demonstrating that the function is unbounded as \(x\) becomes 1.
Without a single, agreed-upon endpoint, the limit \(\lim _{x \rightarrow 1} \frac{1}{x-1}\) must indeed be accounted as nonexistent.

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Most popular questions from this chapter

In Exercises \(73-76,\) find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$\lim _{x \rightarrow \pm \infty} k(x)=1, \lim _{x \rightarrow 1^{-}} k(x)=\infty, \text { and } \lim _{x \rightarrow 1^{+}} k(x)=-\infty$$

Find the average rate of change of the function over the given interval or intervals. \(R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]\)

Find the limits. Are the functions continuous at the point being approached? $$\lim _{x \rightarrow \pi} \sin (x-\sin x)$$

At what points are the functions in Exercises \(13-30\) continuous? $$ y=\frac{1}{x-2}-3 x $$

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