Problem 41 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Analyze the Absolute Value

When approaching from the left (negative side), the absolute value of x can be rewritten. Since we're dealing with values of x that are negative, we have: \[ |x| = -x \] This implies that as \( x \to 0^- \), \( |x| = -x \).

Step 2: Rewrite the Expression

Substitute the expression for \( |x| \) when \( x < 0 \):\[ \lim _{x \rightarrow 0^{-}}rac{1}{x} - rac{1}{-x} \].

Step 3: Simplify the Expression

Since \( rac{1}{-x} = -rac{1}{x} \), substitute this into our expression:\[ \lim _{x \rightarrow 0^{-}} \left( \frac{1}{x} - \left( -\frac{1}{x} \right) \right) \].

Step 4: Combine Terms

Simplify the expression:\[ \lim _{x \rightarrow 0^{-}} \left( \frac{1}{x} + \frac{1}{x} \right) = \lim _{x \rightarrow 0^{-}} \frac{2}{x} \].

Step 5: Evaluate the Limit

As \( x \rightarrow 0^{-} \), the value of \( \frac{2}{x} \) becomes increasingly negative because \( x \) is negative. Consequently, the limit approaches \( -\infty \).

Key Concepts

Absolute Value Function

In mathematics, the absolute value function is denoted by \(|x|\), which represents the distance of a number from zero on a number line, without regard to its direction. This means it takes all numbers and returns their non-negative values.
For example, \(|3| = 3\) and \(|-3| = 3\). This is particularly important in limits, as it helps make distinctions between positive and negative values consistent.

• This becomes crucial when dealing with limits involving left-hand or right-hand values.
• When dealing with negative numbers, \(|x| = -x\), which is a key idea in solving problems where the limit approaches zero negatively.

Understanding how to manipulate the absolute value is essential in understanding the expression involved in your calculus problem. It helps simplify and analyze the situation correctly.

Left-Hand Limit

The concept of a left-hand limit is encountered when analyzing the behavior of a function as it approaches a certain point from the left side or negative direction. It helps in understanding unilateral limits of a function, accounting for behavior when approaching from only one side.

• The notation used is \(\lim_{x \to c^{-}} f(x)\). This is read as “the limit of \f(x)\ as \x\ approaches \c\ from the left.”
• In this context, it ensures that all values considered are less than the approaching point.

When you specifically approach zero from the left, it guarantees you deal only with negative values, which greatly simplifies the expression involved due to the nature of the absolute value function.

Negative Infinity

Negative infinity is a concept representing a value that decreases without bound in the actual number line. In limits, when we say a function approaches negative infinity, it means that as \x\ grows more negative, the function goes lower continuously.
Negative infinity tells you about the direction of divergence.

• In your example, as \(x \to 0^{-}\), the expression involving \(\frac{2}{x}\) shoots down towards immensely negative values.
• This phenomenon suggests that instead of converging to a particular value, the result is an endless descent.

Understanding negative infinity helps in visualizing and correctly solving expressions where outcomes stretch beyond typical numeric constraints.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about reducing them into simpler or more usable forms while retaining equivalence. This involves applying mathematical operations and identities to consolidate terms to reach an easier expression to solve.
The process of simplifying involves a few vital steps:

• Identifying like terms and combining them first.
• Utilizing distribution and factoring techniques if applicable.

For this specific problem:

• The substitution \(\frac{1}{-x} = -\frac{1}{x}\) transformed the expression significantly.
• Combining this with \(\frac{1}{x}\) gives \(\frac{2}{x}\), which highlights the simplification and gives clear insight into its limit behavior.

Mastering simplification not only aids in easing complex calculations but also underpins a stronger understanding of how to approach calculus problems efficiently.