Problem 90 - Thomas Calculus

Question

Use formal definitions to prove the limit statements in Exercises \(89-92 .\) $$\lim _{x \rightarrow 0} \frac{1}{|x|}=\infty$$

Step-by-Step Solution

Answer

The limit is proved to be infinity using the formal definition of limits approaching infinity.

Step 1: Understand the Limit Statement

The statement \(\lim_{x \rightarrow 0} \frac{1}{|x|} = \infty\) implies that as \(x\) approaches 0 (from both the positive and negative sides), \(\frac{1}{|x|}\) grows larger without bound. We aim to formally prove this using the epsilon-delta definition.

Step 2: Set Up the Formal Definition of Limit

To prove \(\lim_{x \rightarrow 0} \frac{1}{|x|} = +\infty\), we need to show that for any given \(M > 0\), there exists a \(\delta > 0\) such that if \(0 < |x| < \delta\), then \(\frac{1}{|x|} > M\).

Step 3: Choose Delta Based on the Given M

Given \( M > 0 \), set \( \delta = \frac{1}{M} \). We need to show that if \( 0 < |x| < \delta \), then \( \frac{1}{|x|} > M \).

Step 4: Verify the Inequality

Assume \( 0 < |x| < \delta = \frac{1}{M} \). Then, \( \frac{1}{|x|} > \frac{1}{\frac{1}{M}} = M \). This confirms that \( \frac{1}{|x|} > M \) when \( 0 < |x| < \frac{1}{M} \).

Step 5: Conclusion

Since for every \(M > 0\), we can find such a \(\delta = \frac{1}{M}\), it follows from the definition of the limit of a function approaching infinity that \(\lim_{x \rightarrow 0} \frac{1}{|x|} = \infty\). Therefore, the statement is proved.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition of a limit is a fundamental concept in calculus used to formally define what it means for a function to have a particular limit at a point. In simple terms, this definition provides a precise mathematical way to say that a function approaches a specific value. To understand this, we start by considering the limit \(lim_{x \rightarrow a} f(x) = L\)\,. This means for any given small distance, called epsilon (\(\varepsilon > 0\)\,), we can find a small distance, called delta (\(\delta > 0\)\,), such that if the input is within this delta distance from \(a\), the output is within the epsilon distance from \(L\)\,.

• This concept is used to ensure that functions get arbitrarily close to a limit value as they approach a specific point.
• For infinite limits, such as \(\lim_{x\rightarrow0}\dfrac{1}{|x|}=\infty\), we use a related definition where instead of epsilon, we look for all \(M > 0\) for the output, ensuring the function output exceeds any large positive number \(M\) as the input gets closer to zero.
• This provides a rigorous structure to visual or intuitive ideas of limits commonly used in calculus.

In the example exercise, this definition is crucial as it helps prove that as \(x\) approaches 0, \(\dfrac{1}{|x|}\) can grow larger without bound. By controlling the input, we can force the output to exceed any level of largeness, which is the essence of the epsilon-delta proof.

Infinite Limits

Infinite limits occur when the value of a function grows endlessly as the input approaches a certain point. In practical terms, this means that as \(x\) approaches a point \(a\), the value of \(f(x)\) can increase to be larger and larger, without any upper limit. Infinite limits are often presented in two types: positive and negative infinity.

• A positive infinite limit \(\lim_{x \rightarrow a} f(x) = +\infty\) indicates the function shoots upwards toward infinity.
• A negative infinite limit \(\lim_{x \rightarrow a} f(x) = -\infty\) indicates the function drops downward without bound.

In the given exercise, when we say \(\lim_{x \rightarrow 0} \frac{1}{|x|} = \infty\), we mean that as \(x\) approaches 0, the expression \(\frac{1}{|x|}\) becomes extremely large. This happens because as the denominator, \(\mid x \mid\), becomes very small, the fraction itself becomes very large. Mathematically, by making \(x\) very close to zero, we don't find an ultimate height that \(\frac{1}{|x|}\) cannot surpass, which is the key feature of an infinite limit.

Limit at Zero

The limit at zero refers to analyzing how a function behaves as the variable \(x\) approaches 0. This concept is particularly important because zero can represent a boundary or point of discontinuity for many functions. Understanding limits at zero helps in evaluating function behavior in contexts such as the origin in graphs.For the function \(\frac{1}{|x|}\), when evaluating \(\lim_{x \rightarrow 0} \frac{1}{|x|} = \infty\), we are interested in how the function behaves when \(x\) is near zero from both the positive and negative sides. Here’s what happens:

• As \(x\) gets closer to zero from the positive side (\(x\rightarrow0^{+}\)), \(\frac{1}{|x|}\) increases without bound.
• Similarly, approaching zero from the negative side (\(x\rightarrow0^{-}\)) also leads \(\frac{1}{|x|}\) to grow infinitely large, because the function takes the absolute value of \(x\), making it positive and small.

This bi-directional analysis is essential as it ensures comprehension that \(\frac{1}{|x|}\) behaves identically, by growing huge, regardless of the side from which zero is approached. This comprehensive consideration of both directions ensures a well-rounded understanding of limits focused at a boundary point like zero.