Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Using the symbols \(M\) and \(N\), give precise definitions of each expression. (a) \(\lim _{x \rightarrow \infty} f(x)=\infty\) (b) \(\lim _{x \rightarrow-\infty} f(x)=\infty\)

Step-by-Step Solution

Answer

(a) For every \( M \), there exists \( N \) such that \( x > N \) implies \( f(x) > M \). (b) For every \( M \), there exists \( N \) such that \( x < N \) implies \( f(x) > M \).

Step 1: Understanding Limits as x Approaches Infinity

The expression \( \lim_{x \rightarrow \infty} f(x) = \infty \) means that as \( x \) increases without bound, \( f(x) \) also increases without bound. In other words, for every real number \( M \), there exists some real number \( N \) such that if \( x > N \), then \( f(x) > M \).

Step 2: Defining the Limit for x Approaching Infinity

For the expression \( \lim_{x \rightarrow \infty} f(x) = \infty \), define it precisely using \( M \) and \( N \): For every positive number \( M \), there exists a corresponding positive number \( N \) such that \( x > N \) implies \( f(x) > M \).

Step 3: Understanding Limits as x Approaches Negative Infinity

The expression \( \lim_{x \rightarrow -\infty} f(x) = \infty \) means that as \( x \) decreases without bound (becomes more and more negative), \( f(x) \) increases without bound. Therefore, for every real number \( M \), there exists a real number \( N \) such that if \( x < N \), then \( f(x) > M \).

Step 4: Defining the Limit for x Approaching Negative Infinity

For the expression \( \lim_{x \rightarrow -\infty} f(x) = \infty \), define it precisely using \( M \) and \( N \): For every positive number \( M \), there exists a corresponding negative number \( N \) such that \( x < N \) implies \( f(x) > M \).

Key Concepts

Limits Approaching Infinity

When we talk about limits approaching infinity, we are observing what happens to a function as the independent variable reaches extremely large or extremely negative values. These are essential concepts in calculus when understanding how functions behave at extreme values. For example:

• For the expression \( \lim_{x \rightarrow \infty} f(x) = \infty \), it indicates that as \( x \) gets larger and larger, \( f(x) \) becomes exceedingly large without any upper limit.
• The expression \( \lim_{x \rightarrow -\infty} f(x) = \infty \), shows that as \( x \) becomes more and more negative, \( f(x) \) still escalates upwards indefinitely.

In these contexts, we involve the symbols \( M \) and \( N \) to rigorously define these situations. Whenever you can pick a profoundly large number \( M \), you can also find a number \( N \) such that \( f(x) \) surpasses \( M \) when \( x \) is larger than \( N \) (or smaller than in the negative infinity context). This method helps to pin down the behavior of a function as it approaches infinity from either direction.

Limits Definition

Limit is a fundamental idea in calculus that describes the behavior of a function as the input approaches a certain value. It helps us to understand function continuity, end behavior, and when functions depend on various conditions. The definition of limits can get precise and mathematical, specifically when talking about infinity.
In calculus:

• The formal definition for \( \lim_{x \to \infty} f(x) = L \), where \( L \) is finite, says for every \( \epsilon > 0 \), there exists an \( N \) such that if \( x > N \), then \( |f(x) - L| < \epsilon \).

In contrast:

• For \( \lim_{x \to \infty} f(x) = \infty \), it’s slightly different: we state that for every real number \( M \), there exists an \( N \) so that for \( x > N \), \( f(x) > M \). It means \( f(x) \) grows bigger than any number we can imagine as \( x \) increases indefinitely.

This formalization helps us apply limits when analyzing functions, ensuring that our conclusions about behavior are accurate and reliable.

Asymptotic Behavior

Asymptotic behavior investigates how a function behaves as it approaches a vast range towards infinity, both positive and negative. This behavior is seen as functions approach lines known as asymptotes, which the function gets close to but never quite touches.
Understanding asymptotic behavior involves:

• Vertical Asymptotes: These occur when a function approaches a certain x-value but becomes unbounded as it nears this x-value.
• Horizontal Asymptotes: Occur when a function approaches a particular y-value as \( x \) heads to infinity or negative infinity.

Sometimes, a function might have oblique or slant asymptotes, usually occurring when the degree of the numerator in a rational function is one more than the degree of the denominator.
This concept relates closely to limits approaching infinity. As \( x \rightarrow \infty \), noting where \( f(x) \) stabilizes or trends toward gives insights into horizontal asymptotes. Recognizing these patterns and relationships is vital in analyzing the long-term behavior of functions.