Problem 43 - Calculus Early Transcendentals

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: Understand the Problem

The task is to find the limit as \( x \) approaches 0 from the left (denoted as \( x \rightarrow 0^{-} \)) for the expression \( \frac{1}{x} - \frac{1}{|x|} \). The absolute value function, \( |x| \), affects this expression differently based on whether \( x \) is negative or positive.

Step 2: Analyze Expressions in Terms of Left-Hand Limit

Since we are considering the limit as \( x \rightarrow 0^{-} \), \( x \) is negative. For negative \( x \), \( |x| = -x \). Consequently, the expression \( \frac{1}{x} - \frac{1}{|x|} \) simplifies to \( \frac{1}{x} - \frac{1}{-x} \).

Step 3: Simplify the Expression

Substitute \( |x| = -x \) into the expression: \[ \frac{1}{x} - \frac{1}{-x} = \frac{1}{x} + \frac{1}{x}. \] This simplifies to \( \frac{2}{x} \).

Step 4: Evaluate the Limit

Evaluate \( \lim_{x \rightarrow 0^{-}} \frac{2}{x} \). As \( x \) approaches 0 from the left, \( x \) is negative and very close to 0, so \( \frac{2}{x} \) becomes \( -\infty \) since the fraction becomes very large and negative.

Key Concepts

Limits and Continuity

Limits are fundamental to calculus, representing the value that a function approaches as the input approaches a certain point. They help us understand the behavior of functions near points of interest, even if the function is not explicitly defined there. In simpler terms, limits allow us to "peek" at what happens as we get infinitely close to a certain value.

Continuity involves both the existence of a limit at a point and the actual value of the function being equal to that limit. A function is continuous at a point if there is no sudden jump or hole at that point.

• If a function is continuous at a point, the limit as you approach that point equals the function's value at that point.
• If there's any interruption, like a hole or jump, the limit may not exist, or if it does, it won't equal the function value there.

Thus, studying limits and continuity helps us understand the seamless behavior—or the lack thereof—of a function across its domain.

One-Sided Limits

One-sided limits focus on the behavior of a function as it approaches a specific point from only one direction—either from the left or the right. These are denoted with a superscript '+' or '-' to indicate the direction.

In the given exercise, we look at the left-handed limit, expressed as \( \lim _{x \rightarrow 0^{-}} \), where \( x \) approaches 0 from the negative side. This means that \( x \) takes on values less than 0 and closes in on 0 from the left direction.

• One-sided limits are crucial when the function behaves differently from the left and right, usually due to a discontinuity at the point.
• For a limit to exist at a point in a standard way, both the left-hand and right-hand limits must converge to the same value.

In our example, examining \( \lim _{x \rightarrow 0^{-}} \frac{2}{x} \), as \( x \) gets closer to 0 from the left, results in the function going towards negative infinity, showcasing a discontinuity.

Absolute Value Function

The absolute value function, denoted \( |x| \), provides the distance of a number \( x \) from zero on a number line, disregarding any sign. It's essential because it allows for expressions and equations involving both negative and positive numbers to be handled smoothly.

In the exercise, the absolute value plays a critical role because it changes how the expression \( \frac{1}{x} - \frac{1}{|x|} \) behaves as \( x \) becomes negative.

• When \( x \) is positive, \( |x| = x \).
• When \( x \) is negative, as it is in \( x \rightarrow 0^{-} \), \( |x| = -x \).

Therefore, when evaluating from the left side, replacing \( |x| \) with \( -x \) results in \( \frac{1}{x} - \frac{1}{-x} = \frac{2}{x} \). This affects the limit's behavior as \( x \) approaches 0 from the left, ultimately leading to the function diverging to negative infinity.