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Chapter 2: Problem 45

Find the limit, if it exists. If the limit does not exist, explain why. $$ \lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{|x|}\right) $$

Short Answer

Expert verified
The limit does not exist because it approaches \(-\infty\).

Step by step solution

01

Understanding the Function

We need to analyze the function inside the limit: \( f(x) = \frac{1}{x} - \frac{1}{|x|} \). Since \(|x|\) denotes the absolute value, for \(x < 0\), we have that \(|x| = -x\). Therefore, \(\frac{1}{|x|} = \frac{1}{-x}\). This means the function simplifies to \( f(x) = \frac{1}{x} + \frac{1}{x} = \frac{2}{x} \) for \(x < 0\).
02

Simplifying the Expression

Using the result from the previous step, the function for \(x < 0\) becomes \( f(x) = \frac{2}{x} \). Now, we need to evaluate the limit \( \lim_{x \to 0^-} \frac{2}{x} \).
03

Approaching the Limit from the Left

As \(x\) approaches 0 from the left (that is, \(x \to 0^-\)), \(x\) is negative and gets closer to zero. The expression \(\frac{2}{x}\) becomes very large in negative terms because dividing a positive number (2) by a very small negative number results in a large negative number. Therefore, \(\lim_{x \to 0^-} \frac{2}{x} = -\infty \).
04

Conclusion on Limit Existence

The limit \(\lim_{x \to 0^-} \frac{1}{x} - \frac{1}{|x|}\) does not exist because the function \( \frac{2}{x} \) approaches \(-\infty\) as \(x\) approaches 0 from the left.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
The absolute value function is a fundamental concept in mathematics, especially in calculus. It provides the non-negative magnitude of a real number without regard to its sign. For any real number \(x\):
  • If \(x\) is positive or zero, \(|x| = x\).
  • If \(x\) is negative, \(|x| = -x\).
In our exercise, we focused particularly on the behavior of \(|x|\) when \(x < 0\). This is because our limit involves approaching zero from the left, where \(x\) is negative. Thus, the absolute value of \(x\) is represented as \(-x\) in this case. This transformation is crucial because it allowed us to simplify the expression inside the limit.
One-Sided Limits
One-sided limits examine the behavior of a function as a variable approaches a specific value from one side, either from the left or the right. In our problem, we dealt with the left-hand limit or the limit as \(x\) approaches 0 from the left.
  • When dealing with left-hand limits, denoted as \(\lim_{x \to c^-}\), \(x\) approaches \(c\) from values less than \(c\).
  • These are valuable in understanding the behavior of functions where there might be discontinuities or piecewise definitions.
For the given function \(\frac{1}{x} - \frac{1}{|x|}\), analyzing the limit from the left was essential because it simplified to a different expression when considered in this region. Understanding one-sided limits helps us focus on the specific behavior of functions at certain critical points.
Limit Does Not Exist
The phrase 'limit does not exist' is used when a function does not converge to a finite number as the independent variable approaches a certain point. This situation can arise in several situations:
  • The function might tend towards infinity.
  • There could be different behavior when approaching the point from the left and the right.
  • The function might oscillate instead of settling towards a particular value.
In our exercise, as \(x\) approached 0 from the left, \(\frac{2}{x}\) tended towards \(-\infty\). This infinitely large negative value signifies that there is no finite number the function approaches; thus, the limit does not exist. This outcome highlights that for limits involving infinite behavior, stating the direction and nature (positive or negative infinity) is crucial.
Algebraic Simplification
Algebraic simplification plays a crucial role in evaluating limits, helping us reduce complex expressions into more manageable forms. This process often involves:
  • Utilizing identities such as absolute value conditions to transform and simplify expressions.
  • Rewriting fractions and terms to reveal their fundamental structure.
  • Simplifying expressions to easily identify trends as variables approach certain values.
In the exercise, simplifying \(\frac{1}{x} - \frac{1}{|x|}\) using the property of the absolute value turned the problem into evaluating \(\frac{2}{x}\) for \(x < 0\). This simplification exposed the behavior of the function as \(x\) approaches zero, showing us clearly that it trends towards an infinite limit, leading us to conclude that the limit does not exist.

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