Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow 0^{-}} \frac{|x|}{x} $$

Step-by-Step Solution

Answer

The limit is \(-1\).

Step 1 - Understand the Expression

The expression given is \( \lim_{x \rightarrow 0^{-}} \frac{|x|}{x} \). This notation indicates that we need to find the limit of \( \frac{|x|}{x} \) as \( x \) approaches 0 from the left (or from negative values).

Step 2 - Consider the Nature of \( |x| \)

For any negative value of \( x \), \( |x| = -x \) since the absolute value of any negative number is its positive counterpart. Hence, for any \( x < 0 \), \( \frac{|x|}{x} = \frac{-x}{x} \).

Step 3 - Simplify the Expression

Given \( \frac{|x|}{x} = \frac{-x}{x} \) for \( x < 0 \), this simplifies to \( -1 \) because \( \frac{-x}{x} = -1 \) as long as \( x eq 0 \).

Step 4 - Determine the Limit

Since \( \frac{|x|}{x} = -1 \) for all \( x < 0 \), as \( x \rightarrow 0^{-} \), the expression is continually \(-1\). Therefore, the limit is \(-1\).

Step 5 - Conclude the Limit

The limit of \( \frac{|x|}{x} \) as \( x \) approaches 0 from the left is \(-1\). This concludes that the function stabilizes at \(-1\) as it approaches the left-hand limit near zero.

Key Concepts

Left-hand limit

In calculus, a left-hand limit refers to the value that a function approaches as the input approaches a certain point from the left, or, more specifically, from the negative direction. To find

• a left-hand limit, we look at what happens to the function as the variable approaches the desired value from values less than the target.
• This means assessing the behavior of the function as the variable gets infinitely close to the point of interest from the left side of the number line.

In our original exercise, the task is to find the left-hand limit of \( \frac{|x|}{x} \) as \( x \) approaches 0. This means that we are considering \( x \) values that are slightly less than 0, such as \( -0.1, -0.01, -0.001\), and so on. By substituting these values into the expression, we find that the expression simplifies to \(-1\), showing stable behavior as \( x \) approaches 0 from the left. Hence, the left-hand limit is \(-1\). This consistent result helps us understand how the function behaves around the point, giving a surface-level insight into the function’s behavior without computing the actual function value at the point.

Absolute value

Absolute value is a fundamental concept in mathematics that measures the magnitude of a number, regardless of its sign. It essentially gives the 'distance' a number is away from zero on a number line, without considering direction. For any real number \( x \):

• If \( x \) is positive, then \( |x| = x\).
• If \( x \) is negative, then \( |x| = -x\).

Absolute value is crucial in our problem of \( \lim_{x \rightarrow 0^{-}} \frac{|x|}{x}\) because as \( x \) approaches 0 from the left (or negative values), \( |x| \) is determined by converting \( x\) to its positive equivalent. When dealing with \( x < 0\), we substitute \( |x| \) with \( -x\). This step is critical because it allows us to simplify the expression \( \frac{|x|}{x}\) to \(-1\), showing the behavior of the function as \( x \) approaches 0 from the left side. The concept of absolute value helps in resolving such expressions, as it requires considering both magnitude and direction.

Negative values approach

When tackling limits that involve approaching a particular point from negative values, it means observing the function’s behavior as the variable comes closer to the target point by taking values that are less than the point itself. For instance, in the provided exercise, we determine the limit of \( \frac{|x|}{x} \) as \( x \) approaches \( 0^{−} \). This notation signifies that \( x \) is approaching 0 from values like \( -0.1, -0.01, \) and further onwards, growing closer to 0 but always staying on the negative side of the number line.

• This approach demands consideration of negative scenarios and how certain operations, such as absolute values, affect them.
• Substituting these negative values into the function during limit evaluation gives insight into the trend or behavior of the function.

In our original exercise, the simplification \( \frac{|x|}{x} = -1\) shows a consistent value as the expression approaches 0 from the left. This negative value approach enables us to determine the left-hand limit, showcasing the stable behavior mapped out by our calculations.