Problem 60 - Precalculus

Question

Use a table of values to evaluate the following limits as \(x\) decreases without bound. Given \(\lim _{x \rightarrow-\infty} \frac{1}{x}=0,\) find the largest negative value of \(x\) such that \(\frac{1}{x} \geq-0.01\)

Step-by-Step Solution

Answer

The largest negative value of \(x\) is \(-100\).

Step 1: Understanding the Problem

To solve the problem, we need to find the value of \(x\) such that \(\frac{1}{x}\) is greater than or equal to \(-0.01\). The limit \(\lim_{x \rightarrow -\infty} \frac{1}{x} = 0\) indicates that as \(x\) becomes more negative, \(\frac{1}{x}\) approaches 0 from the negative side.

Step 2: Setting the Inequality

We want to solve the inequality \(\frac{1}{x} \geq -0.01\). This means finding values for \(x\) where \(x\) is negative and \(\frac{1}{x}\) is at least \(-0.01\).

Step 3: Solve for x

Rearrange the inequality \(\frac{1}{x} \geq -0.01\) to solve for \(x\). We get \(x \leq -100\) by taking the reciprocal of both sides and reversing the inequality sign because we are dealing with reciprocals (and \(x\) is negative).

Step 4: Verify by Test Values

Check the values of \(\frac{1}{x}\) for some test negative values near \(-100\). If \(x = -100\), then \(\frac{1}{x} = -0.01\). If \(x\) is any number less than \(-100\), \(\frac{1}{x}\) will be greater than \(-0.01\). Hence, \(x = -100\) is valid.

Key Concepts

Evaluating Limits

Understanding limits is a fundamental part of calculus. In this context, we are looking at limits as variables approach infinity, particularly negative infinity. Limits help us understand the behavior of functions as they get very large or very small. In our exercise, we focus on evaluating \( \lim_{x \rightarrow -\infty} \frac{1}{x} = 0 \). This tells us that as \( x \) grows more negative, the value of \( \frac{1}{x} \) approaches zero from the negative side. This is a classic example of a limit scenario where the function nears a horizontal asymptote, in this case at \( y = 0 \). Limits are essential for understanding how functions behave in extreme conditions, and they form the foundation for more advanced calculus concepts like continuity and differentiability. When evaluating limits, we often consider not just the value the function approaches but also how it behaves as it gets closer to that value. This insight offers a deeper understanding of the function’s behavior.

Inequalities

Inequalities play a crucial role in determining a range of values that satisfy certain conditions. When working with limits, inequalities often help define when a function behaves in a specific way. In the problem at hand, we address the inequality \( \frac{1}{x} \geq -0.01 \).

• To solve inequalities involving reciprocals with negative values, it's important to remember that multiplying or dividing both sides of an inequality by a negative number requires reversing the inequality sign.
• This is why in our task, the step from \( \frac{1}{x} \geq -0.01 \) to finding \( x \leq -100 \) involves taking the reciprocal of both sides and flipping the inequality sign.

In practice, inequalities guide us in finding a set of values for \( x \) that fulfill the condition given by the function's behavior. Carefully handling inequalities ensures accurate solutions, especially when dealing with limits or any form of function behavior analysis.

Reciprocal Functions

Reciprocal functions are a type of function where each output \( y \) is the multiplicative inverse of the input \( x \). They are represented as \( y = \frac{1}{x} \). Such functions are significant because they exhibit distinct behaviors, particularly as \( x \) approaches both zero and infinity.

• As \( x \) becomes very large or very small, the effect on \( \frac{1}{x} \) shows divergence based on the direction of \( x \). For positive values approaching zero, \( \frac{1}{x} \) tends to positive infinity. For negative values growing larger negatively, as in our exercise, \( \frac{1}{x} \) trends toward zero negatively.
• This behavior helps explain why, despite increasing magnitude of \( x \), \( \frac{1}{x} \) results in a value closer to zero.

Understanding reciprocal functions is pivotal when dealing with limits, especially when setting inequalities to determine ranges of x-values. Grasping how they transform with different x-values can aid in predicting function behavior accurately.