Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=\left\\{\begin{array}{l}g(x), x \geq a, \\ h(x), x

Step-by-Step Solution

Answer

In order for function f to be continuous at x=a, \lim_{x \rightarrow a^-} h(x) must equal to \lim_{x \rightarrow a} g(x).

Step 1: Understand the Definition of Continuity

First, recall definition of continuity. A function f(x) is continuous at a point x = a if \lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a). In other words, the left-hand limit (coming from values less than 'a') equals the right-hand limit (coming from values greater than 'a'), and both of these are equal to the value of the function at that point.

Step 2: Apply the Definition to the Function f

Since f(x) is made up of two functions g and h, apply the definition of continuity at x = a. This gives that the left-hand limit is \lim_{x \rightarrow a^-} h(x) and the right-hand limit (including the point) is \lim_{x \rightarrow a} g(x). For the function f to be continuous at x = a, these two limits should be equal.

Step 3: Formulate the Final Statement

The final statement that makes the function f(x) continuous at x = a is that \lim_{x \rightarrow a^-} h(x) = \lim_{x \rightarrow a} g(x). In other words, as x approaches 'a' from left side, the function h(x) should have the same value as the function g(x) at 'a'. This is what must be true about g and h in order for f to be continuous at x = a.

Key Concepts

Limits

Understanding limits is crucial when examining the behavior of functions as they approach a certain value. A limit, in simple terms, is the value that a function (or sequence) 'approaches' as the input (or index) approaches some value.

For example, if we say that the limit of the function as x approaches c is L, denoted as \( \lim_{x \rightarrow c} f(x) = L \), it means that as x gets closer and closer to c, f(x) gets closer and closer to L.

In the context of piecewise functions, it is vital to consider limits from both directions--from the left (denoted as \( \lim_{x \rightarrow c^-} \) ) and from the right (denoted as \( \lim_{x \rightarrow c^+} \) ). Continuity at a point ensures that these one-sided limits are equal and match the function's value at that point.

Piecewise Functions

Piecewise functions are like a mathematical patchwork: they are defined by different expressions for different intervals of their domain. These functions allow for the representation of complex behaviors that may change rule sets based on the input value.

When dealing with them, one has to look closely at their behavior around the points where the formula changes. These points often require a special focus to ensure that the transition between function segments is seamless, which is integral to function continuity.

An important aspect in the piecewise function is maintaining continuity at these transition points, which necessitates careful consideration of the limits of each piece as we approach this critical junction.

Continuous Functions

Continuous functions can be imagined as a smooth drawing of a curve without lifting the pencil off the paper. Mathematically, a function is said to be continuous at a point if there are no jumps, breaks, or holes at that point.

More formally, a function is continuous at a point (let's say x = a) if and only if three conditions are met: the function is defined at a, the limit exists at a, and the limit of the function as it approaches a is equal to the function's value at a.

Continuous functions exhibit predictable behavior and are easier to analyze and work with, particularly with calculus operations like differentiation and integration.

Limit Definition of Continuity

The limit definition of continuity ties together the concepts of limits and continuous functions. It provides a precise criterion to verify if a function is continuous at a particular point. A function f is continuous at a point x = a if

• The function is defined at x = a, meaning \( f(a) \) exists.
• The limits from both sides of a exist and are equal, meaning \( \lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x)\).
• Furthermore, both one-sided limits equal the function’s value at a, or \( \lim_{x \rightarrow a} f(x) = f(a) \).

Applying this to piecewise functions like in the provided exercise, we ensure a smooth transition by equalizing the left-hand and right-hand limits at the juncture where the function's rule changes.