Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(\lim _{x \rightarrow a} f(x)=L\) and that \(f(a)\) exists (though it may be different from \(L\) ). Prove that \(f\) is bounded on some in- terval containing \(a\); that is, show that there is an interval \((c, d)\) with \(c

Step-by-Step Solution

Answer

Select \(\epsilon = 1\) for the limit definition; for \(|f(x) - L| < 1\), set \(M = \max(|L - 1|, |L + 1|)\) on \((a-\delta, a+\delta)\).

Step 1: Definition of Limit

We know that \[ \lim_{x \to a} f(x) = L \]means for every \( \epsilon > 0 \), there is a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \(|f(x) - L| < \epsilon\).

Step 2: Choose Epsilon

Select \( \epsilon = 1 \), which is often a straightforward choice to find a bound. According to the definition of the limit, there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \(|f(x) - L| < 1\).

Step 3: Establish Boundaries for the Interval

Consider the interval \((a - \delta, a + \delta)\). For every \( x \) in this interval, since \(|f(x) - L| < 1\), it follows that \[ -1 < f(x) - L < 1 \]or equivalently, \[ L - 1 < f(x) < L + 1 \].

Step 4: Define a Bound M

Since \(f(x)\) is bounded by \(L - 1\) and \(L + 1\) in \((a - \delta, a + \delta)\), we can set\[ M = \max(|L - 1|, |L + 1|). \]Thus, for all \(x\) in \((a - \delta, a + \delta)\), \[ |f(x)| \leq M. \]

Step 5: Conclusion

Therefore, we have found an interval \((c, d) = (a - \delta, a + \delta)\) and a constant \(M\), such that \(|f(x)| \leq M\) for all \(x\) in this interval. Thus, the function \(f\) is bounded on \((c, d)\).

Key Concepts

Definition of Limit

The concept of a limit is fundamental in calculus and helps us understand the behavior of functions as they approach a certain point. When we say that \( \lim_{x \to a} f(x) = L \), we are describing how the function \( f(x) \) behaves as \( x \) gets closer and closer to a particular value \( a \).
The formal definition states that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if the distance between \( x \) and \( a \) is less than \( \delta \) but not equal to zero, then the distance between \( f(x) \) and \( L \) is less than \( \epsilon \).
This is expressed as:

• \( 0 < |x - a| < \delta \) implies \( |f(x) - L| < \epsilon \).

Limits allow us to analyze the behavior of functions when direct substitution is not possible, and they form the groundwork for more complex calculus concepts such as continuity and differentiation.

Bounded Function

A bounded function is one that does not exceed a certain value within a given range. Specifically, for a function \( f(x) \), if there is a constant \( M \) such that \( |f(x)| \leq M \) for all \( x \) in a certain interval, then \( f(x) \) is said to be bounded on that interval.
In our exercise, we are tasked with showing that if a function \( f \) approaches a limit \( L \) as \( x \) approaches \( a \), then \( f \) must be bounded in some interval around \( a \). We use the definition of the limit to find a \( \delta \) such that \( |f(x) - L| < \epsilon \).
By choosing \( \epsilon = 1 \), we conclude that the function's values are trapped between \( L - 1 \) and \( L + 1 \), hence \( f(x) \) is bounded by \( M = \max(|L - 1|, |L + 1|) \) within the interval \( (a - \delta, a + \delta) \).
Thus, the function does not "escape" to infinity within this interval, providing us with a practical way to manage the behavior of \( f(x) \). This ensures that the function behaves predictably and stays within reasonable limits near \( a \).

Intervals in Calculus

Intervals are fundamental in calculus as they help us define and understand the range over which functions behave in a certain manner. They can be open, closed, or half-open, denoting whether endpoints are included in the interval.
An interval like \( (c, d) \) refers to all points \( x \) such that \( c < x < d \). This open interval does not include the endpoints \( c \) or \( d \).
In the context of limits, intervals are used to express the neighborhood around a point \( a \) where certain conditions, like bounding \( f(x) \), are satisfied. By defining an interval \( (a-\delta, a+\delta) \), we instantly know where these conditions apply.

• Open intervals do not include their endpoints, making them useful for expressing behaviors close to but not at the endpoint.
• Closed intervals, written as \( [c, d] \), include both endpoints.
• Half-open intervals include one endpoint but not the other, such as \( [c, d) \).

Using intervals, we can effectively explore and define the segments of a function’s domain where important characteristics like being bounded apply, ensuring clear understanding and control over the function's behavior.