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 interval containing \(a\); that is, show that there is an interval \((c, d)\) with \(c

Step-by-Step Solution

Answer

The function \( f \) is bounded by \(|L| + 1\) on the interval \((a-\delta, a+\delta)\).

Step 1: Understanding the Given Information

We know that the limit \( \lim_{x \rightarrow a} f(x) = L \) and that \( f(a) \) exists. Although \( f(a) \) may not equal \( L \), the function approaches \( L \) as \( x \) approaches \( a \).

Step 2: Applying the Definition of Limit

By the definition of a limit, for any \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \). This means \( f(x) \) is very close to \( L \) in the interval \( (a - \delta, a + \delta) \) except possibly at \( a \).

Step 3: Choosing an Appropriate \( \epsilon \)

Choose \( \epsilon = 1 \). Therefore, there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < 1 \). This implies that \( L - 1 < f(x) < L + 1 \).

Step 4: Defining the Bounded Interval

The interval \((c, d)\) can be defined as \((a-\delta, a+\delta)\). By our choice of \( \epsilon \), we know that \(|f(x)| < |L| + 1\) wherever \(0 < |x-a| < \delta\).

Step 5: Bounding the Function on the Interval

Since \(|f(x)|\) is bounded by \(|L| + 1\) in the interval \((a-\delta, a+\delta)\), the function is indeed bounded in this interval. Hence, \( |f(x)| \leq M = |L| + 1 \) for all \( x \) in \((a-\delta, a+\delta)\).

Key Concepts

Definition of limit

In calculus, the definition of a limit describes the behavior of a function as it approaches a particular point. When we say that \( \lim_{x \rightarrow a} f(x) = L \), it means that as \( x \) gets closer and closer to \( a \), the function \( f(x) \) gets closer to the value \( L \). Importantly, the concept of a limit does not require \( f(x) \) to actually equal \( L \) at \( x = a \), or even that \( f(a) \) exists or equals \( L \). This is because the limit concerns how \( f(x) \) behaves near \( a \), not exactly at \( a \).

For many functions, knowing about limits allows us to predict the function's behavior very precisely. In practice, it means we can talk about the value a function is approaching. Limits are foundational in calculus as they lay the groundwork for more complex topics like continuity and derivatives.

Bounded function

A function is considered bounded on a particular interval if there is a real number \( M \) such that the absolute value of the function is always less than or equal to \( M \) for any point in that interval. In simpler terms, a bounded function does not go to infinity and stays within a certain range on the specified interval.

To visualize it, imagine a function that oscillates up and down. If we can place a horizontal band around the graph of the function within a certain interval, and the graph never leaves that band, then the function is bounded in that interval.

• A bounded function guarantees that the values remain within some minimum and maximum.
• This concept is useful in proving certain properties, like continuity, as it prevents the function from having infinite oscillation near a point.

Epsilon-delta definition

The epsilon-delta definition is a formal way of defining what it means for a limit to exist. This definition uses two numbers: \( \epsilon \) and \( \delta \).

Here's how it breaks down:

• We choose a very small number \( \epsilon > 0 \). Think of \( \epsilon \) as a tiny window around \( L \), the value we believe our function approaches.
• We then find a \( \delta > 0 \) such that whenever \( x \) is within \( \delta \) distance from \( a \), the function \( f(x) \) falls within the \( \epsilon \) window around \( L \). This is represented as: \( 0 < |x - a| < \delta \) implies \( |f(x) - L| < \epsilon \).

This process ensures that we can make \( f(x) \) as close as we like to \( L \) by choosing \( x \) sufficiently close to \( a \). It is a precise mathematical way of describing the intuitive approachability of \( f(x) \) towards \( L \) as \( x \) approaches \( a \). This foundation is crucial for understanding more advanced concepts in calculus and mathematical analysis.