91Ó°ÊÓ

The epsilon-delta definition is a fundamental concept in calculus, used to rigorously define the limit of a function. In simple terms, it provides a way to express that a function can be made to get as close as desired to a particular value (the limit) by adjusting the inputs within a certain range.

When we say a function \( f(x) \) approaches a limit \( L \) as \( x \) approaches \( c \), we are saying that for every positive number \( \varepsilon \), there exists another number, \( \delta > 0 \), such that

• \( |f(x) - L| < \varepsilon \)
• whenever \( 0 < |x - c| < \delta \)

This means that \( f(x) \) can be kept within an interval of \( L \) by keeping \( x \) within an interval of \( c \). The idea is to make the function's output as close as needed to \( L \) by narrowing the range of \( x \).

The epsilon-delta definition is crucial for understanding the behavior of functions at points where they may not be straightforward to evaluate, allowing for a rigorous proof of limits.

Absolute value inequalities like \( |a| < b \) express the idea that the absolute value of \( a \) is less than \( b \). In the context of limit problems, this often translates into ensuring the difference between the function value \( f(x) \) and the limit \( L \) is less than a small number, \( \varepsilon \).

The inequality \( |-3(x + 1)| < 0.03 \) states that the expression inside the absolute value, regardless of its sign, should be smaller than 0.03. By simplifying, we express it as

• \( |x + 1| < \frac{0.03}{3} \), or \( |x + 1| < 0.01 \)

This process of handling the absolute values is critical as it helps isolate \( x \) to determine \( \delta \). The inequality shows us how tightly \( x \) must cluster around \( c \) to maintain the desired \( \varepsilon \) closeness.

Finding \( \delta \) boils down to expressing how narrowly \( x \) can vary around \( c \) to keep \( f(x) \) near \( L \). This is achieved by rearranging the absolute value inequality to isolate \( x \).

In our example, the inequality leads us from \( |-3(x + 1)| < 0.03 \) to \( |x + 1| < 0.01 \). This directly translates to needing \( 0 < |x + 1| < 0.01 \) to achieve the requirement of \( |f(x) - L| < \varepsilon \). Thus, we conclude:

• \( \delta = 0.01 \)

Displaying the interval \( (c - \delta, c + \delta) \) assures us of the correctness. Finding \( \delta \) is pivotal because it determines the bounds within which \( x \) must lie to meet the conditions set by \( \varepsilon \). This concept ensures continuity and reliability in the function's behavior near \( x = c \).

Continuity of functions hinges on the ability for a function's graph to be drawn without lifting the pencil from paper at point \( c \). It's where limits, function values, and continuity criteria (

• \( f(c) \) is defined
• \( \lim_{x \to c} f(x) = f(c) \)

) are all met.

In our particular situation, the function \( f(x) = -3x - 2 \) is linear, which inherently meets these criteria. Thus for any point \( c \), including \( c = -1 \), the function is continuous. This means:

• There are no jumps, breaks, or discontinuities at this point
• The limit directly equals the function's value at \( c \)

Understanding continuity assures us we can apply limits reliably, knowing they'll give valid function evaluations, crucial for calculus operations.