91Ó°ÊÓ

The Squeeze Theorem is a handy tool for evaluating limits and is particularly useful when dealing with functions that might behave unpredictably. The theorem states:

• If you have three functions, say \( f(x) \), \( g(x) \), and \( h(x) \), and for all \( x \) in a certain interval except possibly at a point \( c \), \( f(x) \leq g(x) \leq h(x) \).
• And if \( \lim_{x \to c} f(x) = L = \lim_{x \to c} h(x) \), then \( \lim_{x \to c} g(x) = L \) as well.

This means that if a function is trapped between two other functions that have the same limit at a point, then the squeezed function must also have that same limit. It's like pinning down the function between two boundaries that converge at a point.
In our exercise, the Squeeze Theorem helps show that \( \lim_{x \to 0} xg(x) = 0 \) by comparing \( |xg(x)| \) with \(|x|M\), which approaches zero as \( x \to 0 \). This establishes the continuity at \( 0 \).

A bounded function is a function that does not "go to infinity" either positively or negatively, within a certain interval. In simpler terms, there is a limit to how large or how small the function values can get. For example, a function defined on

• an interval \([a, b]\), is bounded if there exists a number \( M \) such that for every \( x \) in the interval, the function satisfies \(|g(x)| \leq M\).

In the exercise, the function \( g(x) \) is bounded on the interval \([0, 1]\), signaling that there exists a constant \( M \) which serves as an upper bound on \(|g(x)|\).
This characteristic is crucial because it allows the expression \( xg(x) \) to also be bounded, ensuring that \( \lim_{x \to 0} xg(x) \) can be more easily evaluated using the Squeeze Theorem. By knowing \( g(x) \) is bounded, it aids us in establishing that \( |xg(x)| \) will always be less than or equal to \( |x|M \), effectively controlling its behavior as \( x \) approaches zero.

The concept of a limit deals with how a function behaves as the input approaches a certain value. It helps us understand behavior at points where the function might not be exactly defined, or reveal insights about the trend of values within a small area.
Mathematically, the limit \( \lim_{x \to c} f(x) = L \) illustrates that as \( x \) gets closer to \( c \), the output \( f(x) \) gets closer to \( L \).

• In other words, for every small positive number \( \epsilon \), there exists a \( \delta \) such that whenever \( 0 < |x-c| < \delta \), it follows that \(|f(x) - L| < \epsilon\).

In practical terms, evaluating limits is fundamental to ensuring the continuity of functions because continuity at a point implies the limit as \( x \) approaches that point equals the function's value at the point.
In this scenario, the exercise seeks to determine that \( \lim_{x \to 0} xg(x) = 0 \), which speaks to the function \( xg(x) \) approaching a value of 0 precisely as \( x \) zeroes in on the point zero. This limit's evaluation is integral to confirming that \( x \mapsto xg(x) \) remains continuous at \( x = 0 \).