Problem 67 - Calculus Early Transcendentals

Question

$$ \lim _{x \rightarrow-\infty} \frac{1}{x}=0 $$

Step-by-Step Solution

Answer

As \( x \rightarrow -\infty \), \( \frac{1}{x} \rightarrow 0 \).

Step 1: Understanding the Limit Definition

The expression \( \lim_{x \rightarrow -\infty} \frac{1}{x} \) tells us to find the value that \( \frac{1}{x} \) approaches as \( x \) becomes very large negatively.

Step 2: Analyzing the Function

Consider the function \( f(x) = \frac{1}{x} \). As \( x \) becomes very large negatively (i.e., \( x \rightarrow -\infty \)), the denominator \( x \) in the fraction \( \frac{1}{x} \) also becomes large negatively.

Step 3: Observing the Behavior of the Fraction

As \( x \) is negative and its magnitude is increasing, the absolute value of the fraction \( \frac{1}{x} \) gets smaller. For example, when \( x = -10 \), \( \frac{1}{x} = -0.1 \); when \( x = -100 \), \( \frac{1}{x} = -0.01 \); so, as \( x \rightarrow -\infty \), \( \frac{1}{x} \rightarrow -0 \).

Step 4: Conclusion

We conclude that the limit of \( \frac{1}{x} \) as \( x \) approaches negative infinity is 0 because the fraction becomes closer to zero and negative values tend to approach zero from the negative side.

Key Concepts

Infinite Limits

Infinite limits occur when a function's value grows without bound as the input grows larger indefinitely. In calculus, we often deal with scenarios where the input either becomes infinitely large or infinitely small.

A distinct characteristic of infinite limits is the constant increase or decrease of the output without reaching a specific boundary or value. For example, consider the function defined by \(rac{1}{x}\). As \(x\) approaches infinity or negative infinity, this function demonstrates a trend towards a particular value, in this case, zero.

Understanding infinite limits is crucial for analyzing how functions behave at extreme values. In practical terms, these limits help predict whether a graph will shoot upwards towards infinity or decrease downwards towards negative infinity as the independent variable, often time or another specified input, progresses beyond normal operational constraints.

Behavior of Rational Functions

Rational functions, which are the ratio of two polynomials, have complex behaviors especially when inputs become extremely large or small.

Analyzing rational functions requires studying both the numerator and denominator independently. When considering limits such as \(\lim_{x \to -\infty} \frac{1}{x}\), the behavior significantly depends on the balance between the growth rates of these two polynomial expressions.

A key aspect of rational functions' behavior is their asymptotic behavior. For example, \(\frac{1}{x}\) illustrates that as \(x\) becomes larger negatively, the fraction's value approaches zero. Here, zero acts as a horizontal asymptote—a line that the graph of the function gets indefinitely closer to but never actually touches.

Approaching Negative Infinity

Approaching negative infinity is a central idea in exploring limits where values of interest are moving towards very large negative numbers. This process involves continuously decreasing values of \(x\).

For instance, with the limit \(\lim_{x \rightarrow -\infty} \frac{1}{x}\), as \(x\) decreases further into the negatives, the magnitude of numbers increases, making values like \(-10, -100, -1000\) relevant. Each step shows how \(\frac{1}{x}\) becomes more trivial, shrinking toward zero.

Understanding why -0 is mentioned adds depth to this concept. Although \(-0\) and \(0\) are mathematically equivalent, saying approaches from the negative reminds us of the direction from which the limit is approaching zero. Approaching negative infinity highlights how function behaviors can be direction-sensitive, showing us not just where they are heading, but also how they are getting there.