Problem 70 - Calculus Early Transcendentals

Question

$$ \begin{array}{c}{\text { Formulate a precise definition of }} \\ {\quad \lim _{x \rightarrow-\infty} f(x)=-\infty} \\ {\text { Then use your definition to prove that }} \\ {\lim _{x \rightarrow-\infty}\left(1+x^{3}\right)=-\infty}\end{array} $$

Step-by-Step Solution

Answer

The limit of \( 1 + x^3 \) as \( x \to -\infty \) is \( -\infty \) as it decreases without bound.

Step 1: Understanding the Limit Definition

To define \( \lim_{x \rightarrow -\infty} f(x) = -\infty \), we need to understand that for every real number \( M \), there exists a real number \( N \) such that if \( x < N \), then \( f(x) < M \). This means as \( x \) decreases without bound, \( f(x) \) will also decrease without bound.

Step 2: Working with the Function

Consider the function \( f(x) = 1 + x^3 \). We need to show that \( \lim_{x \rightarrow -\infty} (1 + x^3) = -\infty \) using the limit definition.

Step 3: Setting up the Proof

We start by fixing an arbitrary real number \( M \). According to our definition, we need to find some real \( N \) such that for every \( x < N \), we have \( 1 + x^3 < M \).

Step 4: Solving the Inequality

From \( 1 + x^3 < M \), we rearrange to get \( x^3 < M - 1 \). Since \( x^3 \) is a monotonic function decreasing as \( x \) becomes strongly negative, we solve for \( x \) to get \( x < (M - 1)^{1/3} \).

Step 5: Choosing the Bound for \( x \)

Choose \( N = (M - 1)^{1/3} \). For every \( x < N \), the condition \( x^3 < M - 1 \) holds, hence \( 1 + x^3 < M \).

Step 6: Conclusion of the Proof

Thus, for any real number \( M \), there exists an \( N = (M - 1)^{1/3} \) such that for all \( x < N \), \( 1 + x^3 < M \). Hence, by definition, \( \lim_{x \rightarrow -\infty}(1 + x^3) = -\infty \).

Key Concepts

Definition of Limits at Negative Infinity

Understanding limits, especially as they approach negative infinity, is essential for comprehending how functions behave as the input values get smaller and smaller. A limit at negative infinity essentially describes the behavior of a function as its input decreases without bound.
In layman's terms, when we say that the limit of a function \( f(x) \) as \( x \) approaches negative infinity is equal to negative infinity, written as \( \lim_{x \rightarrow -\infty} f(x) = -\infty \), we mean:

• For every real number \( M \), no matter how large or small, there will always be a point \( x \) beyond which all function values \( f(x) \) are less than this number \( M \).
• This signifies that as \( x \) becomes larger and negative, \( f(x) \) will continue to decrease past any given bound.

This concept is critical in calculus because it helps predict and understand the behavior of curves without graphing them at extreme negative values of \( x \).

Proof of Limits at Negative Infinity

The proof of a limit statement involves showing that a particular condition is met for all sufficiently negative numbers. The process involves:

• Selecting an arbitrary real number \( M \). This number represents any lower limit we want \( f(x) \) to be less than.
• Finding a corresponding real number \( N \) such that for all \( x < N \), the function \( f(x) \) satisfies the condition \( f(x) < M \).

To prove that \( \lim_{x \rightarrow -\infty}(1 + x^3) = -\infty \):

• Start with the inequality \( 1 + x^3 < M \), which leads to \( x^3 < M - 1 \).
• As \( x^3 \) is monotonic decreasing (further explained later), this is true for all \( x \) less than \((M-1)^{1/3}\).
• Thus, \( N = (M-1)^{1/3} \) will suffice for \( x < N \), thereby proving that \( 1 + x^3 \) decreases past any \( M \).

The proof involves algebraic manipulation and understanding function behavior at extreme values. Successfully proving it solidifies the definition and understanding of limits at negative infinity.

Monotonic Functions

Monotonic functions are functions whose outputs move in a single direction as their inputs increase or decrease. Specifically, a function can be either increasing or decreasing:

• An increasing function has outputs that rise as inputs grow.
• A decreasing function has outputs that drop as inputs shrink or grow negatively.

For the function \( f(x) = x^3 \), which is used in our exercise, we recognize it as a monotonic decreasing function when \( x \) goes to negative infinity. This is because:

• The cube function \( x^3 \) continues to become more negative as \( x \) decreases beyond zero into negative territory.
• This reliable behavior allows us to predict that \( f(x) = x^3 \) reaches and surpasses any negative limit \( M \) if we go far enough in the negative direction.

Understanding monotonic functions aids in proving limit behaviors, as knowing the direction of the function's growth is pivotal. This simplifies showing \( N \) exists such that every input less than it results in outputs smaller than a set boundary \( M \).