Problem 71 - Calculus: Early Transcendentals

Question

70-72. Proving that \(\lim _{x \rightarrow a} f(x) \neq L\) Use the following definition for the nonexistence of a limit. Assume \(f\) is defined for all x near a, except possibly at a. We write \(\lim f(x) \neq L\) if for some \(\varepsilon>0,\) there is no value of \(\delta>0\) satisfying the condition $$|f(x)-L|<\varepsilon \text { whenever } 0<|x-a|<\delta$$ Prove that \(\lim _{x \rightarrow 0} \frac{|x|}{x}\) does not exist.

Step-by-Step Solution

Answer

Explain your answer. **Answer:** No, the limit of the function \(\lim_{x \rightarrow 0} \frac{|x|}{x}\) does not exist. This can be demonstrated by observing that the function behaves differently when x approaches 0 from the positive side (\(\lim_{x \rightarrow 0^{+}} f(x) = 1\)) and the negative side (\(\lim_{x \rightarrow 0^{-}} f(x) = -1\)). Since the two-sided limits are not equal, the overall limit does not exist. Furthermore, using the given definition for nonexistence of a limit, we can show that there is an \(\varepsilon = \frac{1}{2}\) for which no \(\delta > 0\) satisfies the condition \(|f(x) - L| < \varepsilon\) when \(0 < |x| < \delta\), proving the limit does not exist.

Step 1: Express f(x) as a piecewise function

Intuitively, we can see that the function \(f(x) = \frac{|x|}{x}\) behaves differently for positive and negative values of x. Let's rewrite this function as a piecewise function to analyze it better: $$ f(x) = \frac{|x|}{x} = \begin{cases} 1, & \text{if } x>0 \\ -1, & \text{if } x<0 \end{cases} $$ Now we can analyze the behavior of the function as x approaches 0 from both the positive and negative sides.

Step 2: Determine the function's behavior as x approaches 0 from positive side

Let's consider the behavior of the function as x approaches 0 from the positive side (i.e., \(x \rightarrow 0^{+}\)). When x is positive, the function f(x) takes the value 1, so the limit as x approaches 0 from the positive side will be: $$ \lim_{x \rightarrow 0^{+}} f(x) = 1 $$

Step 3: Determine the function's behavior as x approaches 0 from the negative side

Now, let's consider the behavior of the function as x approaches 0 from the negative side (i.e., \(x \rightarrow 0^{-}\)). When x is negative, the function f(x) takes the value -1, so the limit as x approaches 0 from the negative side will be: $$ \lim_{x \rightarrow 0^{-}} f(x) = -1 $$

Step 4: Compare the two-sided limits

To determine whether the limit exists, we need to check if the limits as x approaches 0 from both the positive and the negative side are equal: $$ \lim_{x \rightarrow 0^{+}} f(x) \neq \lim_{x \rightarrow 0^{-}} f(x) $$ Since the limits from both positive and negative sides are not equal, this implies that the limit of f(x) as x approaches 0 does not exist according to the two-sided limit test.

Step 5: Proving nonexistence of the limit using the given definition

Now we need to prove this using the given definition for nonexistence of a limit. Let's assume that the limit L exists. We will show that there exists an \(\varepsilon > 0\) such that no \(\delta > 0\) can satisfy the condition \(|f(x) - L| < \varepsilon\) when \(0 < |x| < \delta\). Let's choose \(\varepsilon = \frac{1}{2}\). If the limit L exists, then for any \(0 < |x| < \delta\), we should have the condition \(|f(x) - L| < \frac{1}{2}\). However, we've already established that the function behaves differently when x is positive and when x is negative, meaning that it can't satisfy the condition for all x near 0. Therefore, we've shown that there is an \(\varepsilon > 0\) for which no \(\delta > 0\) satisfies the given condition, and so the limit of f(x) as x approaches 0 does not exist.

Key Concepts

Understanding Piecewise Functions

Piecewise functions are mathematical expressions defined by different formulas for different intervals of the variable. Unlike standard functions which have a single rule applying to all values in their domain, piecewise functions can behave differently depending on the input value.

Think of a piecewise function as a chameleon that changes its appearance based on its surroundings. For example, consider a function that serves different dishes for lunch and dinner. Similarly, a piecewise function could have one expression for positive inputs and another for negative inputs. Illustrating this through an example:

\[f(x) = \begin{cases}1, & \text{if } x>0 \-1, & \text{if } x<0\end{cases}\]

Here, the function has two 'pieces', one for when x is positive and another for when x is negative. Understanding piecewise functions is crucial because they can model complex, real-world situations where the relationship between variables is not uniform across their entire range.

Exploring Two-Sided Limits

Two-sided limits are essential in understanding the behavior of functions as the input approaches a certain value from both sides. In this context, a two-sided limit asks the question, 'What value does the function approach as we get close to a specific point from both directions?'

For a limit to exist at a certain point, the function must approach the same value from both the left and the right. This concept is similar to approaching a street intersection from two opposite directions; to meet at the same point, both paths must converge to the intersection. Mathematically, this is expressed as:

\[\lim_{x \rightarrow c^+} f(x) = \lim_{x \rightarrow c^-} f(x)\]

where the superscripts '+' and '-' denote the right-hand and left-hand limits, respectively. If these two limits do not agree, as can happen with piecewise functions, the overall limit at that point does not exist.

Behavior of Functions Near a Point

The behavior of functions near a point is intriguing, as it often unveils the function's nature within its domain, especially around discontinuities or points where the function behaves erratically. To fully understand the function's behavior near a point, it is often helpful to consider not only the function's value precisely at the point but also how it behaves in an arbitrarily small neighborhood around it.

Imagine standing on a hilltop and trying to predict the landscape in your immediate vicinity based on your current location. Similarly, by analyzing how a function behaves as the variable approaches a particular point (but does not actually reach it), we can deduce the function's continuity, possible discontinuities, or even asymptotic behavior.

The mathematical process for examining the behavior at a point involves taking limits, which serve as a magnifying lens allowing us to zoom in on the function's behavior without being affected by the value at the point itself, especially if the function is undefined at that point.