Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give the appropriate \(\varepsilon-\delta\) definition of each statement. $$ \lim _{x \rightarrow c^{-}} f(x)=L $$

Step-by-Step Solution

Answer

For all \(\varepsilon > 0\), \(\exists \delta > 0\) s.t. if \(c - \delta < x < c\), then \(|f(x) - L| < \varepsilon\).

Step 1: Define the \\varepsilon-\\delta Concept

The \(\varepsilon-\delta\) definition of a left-hand limit states that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(c - \delta < x < c\), then \(|f(x) - L| < \varepsilon\). This helps formalize the notion that \(f(x)\) approaches \(L\) as \(x\) gets arbitrarily close to \(c\) from the left side.

Step 2: Identify the Components

In the problem statement, \(L\) is the limit value that \(f(x)\) approaches, and \(c\) represents the point from which \(x\) approaches from the left. The key is to ensure \(|f(x) - L| < \varepsilon\) whenever \(x\) is in the interval \(c - \delta < x < c\).

Step 3: Write the \\varepsilon-\\delta Definition

Thus, the \(\varepsilon-\delta\) definition of \(\lim_{x \to c^{-}} f(x) = L\) is: For every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(c - \delta < x < c\), then \(|f(x) - L| < \varepsilon\). This formal statement encapsulates the idea that as \(x\) approaches \(c\) from the left, \(f(x)\) approaches \(L\).

Key Concepts

Left-Hand Limit

When we talk about the left-hand limit of a function, we are focusing on how the function behaves as the input values get very close to a certain point from the left side. Imagine you are on a number line approaching a point, but only moving from the left side. That’s exactly what we consider in left-hand limits.
In mathematical terms, if we say \( \lim_{x \to c^{-}} f(x) = L \), it means that as \( x \) gets closer to \( c \) from values less than \( c \), the function \( f(x) \) gets closer to \( L \).

• "\( c^{-} \)" signifies we are looking at the direction from the left.
• "\( f(x) \to L \)" indicates that the function's values are getting nearer to \( L \).

Understanding left-hand limits helps us dissect how functions behave near certain points, particularly when they are not defined exactly at those points.

Limit of a Function

The limit of a function is a fundamental concept in calculus that explains how a function behaves as its input approaches a certain value.
If we say \( \lim_{x \to c} f(x) = L \), we mean that as \( x \) gets closer and closer to \( c \), \( f(x) \) gets closer to \( L \). It doesn't matter what happens at \( c \); what matters is the behavior of the function as \( x \) approaches \( c \).

• The limit tells us about the function’s trend as \( x \) is near \( c \).
• We can have different types of limits, including one-sided limits like the left-hand and right-hand limits.
• Understanding limits helps us make sense of functions at points we can't explicitly evaluate them.

Limits allow us to explore continuity, differentials, and integrals in calculus.

Mathematical Notation

Mathematical notation is a universal language used by mathematicians to express concepts efficiently and accurately. In calculus, and indeed many areas of mathematics, precise notation is key to conveying complex ideas in a straightforward manner.
For example, the notation \( \varepsilon-\delta \) is used to describe the formal approach to limits. Here’s a quick breakdown:

• \( \varepsilon \) (epsilon): Represents any small positive number, showing how close \( f(x) \) needs to be to the limit \( L \).
• \( \delta \) (delta): Indicates the range within which \( x \) needs to be from \( c \) to maintain \( |f(x) - L| < \varepsilon \).

Mathematical notation like this makes it possible to express intricate concepts such as limits with clarity and brevity. It is a powerful tool that turns a potentially complex narrative into a concise statement, allowing us to tackle problems methodically.