Problem 14 - Calculus Early Transcendentals

Question

\(13-14\) Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. \(g(\mathrm{x})=2 \sqrt{3-\mathrm{x}}, \quad(-\infty, 3]\)

Step-by-Step Solution

Answer

The function \(g(x) = 2\sqrt{3-x}\) is continuous on \((-\infty, 3]\).

Step 1: Understand the definition of continuity

A function is continuous at a point \(c\) if \(\lim_{x \to c} f(x) = f(c)\). This means that as \(x\) approaches \(c\), the value of \(f(x)\) approaches \(f(c)\). For a function to be continuous on an interval, it must be continuous at every point within the interval.

Step 2: Define the function and the interval

The given function is \(g(x) = 2\sqrt{3-x}\), and we need to show it's continuous on the interval \((-\infty, 3]\). The interval means we need to consider all points \(x\leq 3\).

Step 3: Calculate the limit as x approaches a point c where c < 3

For a point \(c\) where \(c < 3\), the function is defined, and since it's composed of elementary functions, it's continuous. Thus, \(\lim_{x \to c} 2\sqrt{3-x} = 2\sqrt{3-c}\), which equals \(g(c)\).

Step 4: Check the limit as x approaches 3 from the left

Let's find \(\lim_{x \to 3^-} 2\sqrt{3-x}\). As \(x\) approaches 3 from the left, \(3-x\) approaches 0 but stays positive, meaning \(2\sqrt{3-x}\) approaches 0. Thus, \(\lim_{x \to 3^-} g(x) = 0\).

Step 5: Evaluate g(3) directly

Calculate \(g(3) = 2\sqrt{3-3} = 2\times 0 = 0\).

Step 6: Compare the limit and the function value at x=3

Since \(\lim_{x \to 3^-} g(x) = g(3) = 0\), the function is continuous at \(x=3\).

Step 7: Conclude that g(x) is continuous on the interval

Since the function \(g(x)\) is continuous at \(x=3\) and for all \(x < 3\) in the given interval due to the properties of limits and its definition, we conclude that \(g(x)\) is continuous on \((-\infty, 3]\).

Key Concepts

Definition of Continuity

To start with, the concept of continuity is fundamental in calculus. A function is considered continuous at a point if its graph is unbroken at that point. More formally, we say a function, \( f(x) \), is continuous at a point \( c \) if the limit of \( f(x) \) as \( x \) approaches \( c \) equals \( f(c) \).
This can be mathematically expressed as:

• The limit \( \lim_{x \to c} f(x) = f(c) \).

If a function is continuous at every point in an interval, we say it is continuous on that interval. Continuity ensures there are no gaps, jumps, or holes at any point \( c \) within the domain. To assess continuity, check if the limit at each point matches the function's actual value at that point.

Properties of Limits

The properties of limits are crucial in analyzing continuity. They provide the tools to evaluate the behavior of functions as \( x \) approaches specific points. Here are some essential properties:

• Constant Rule: The limit of a constant is the constant itself. \( \lim_{x \to c} k = k \).
• Sum Rule: The limit of a sum is the sum of the limits. \( \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \).
• Product Rule: The limit of a product is the product of the limits. \( \lim_{x \to c} [f(x)g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \).
• Quotient Rule: The limit of a quotient is the quotient of the limits, provided the denominator is not zero. \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \), \(g(x) eq 0 \).
• Root Rule: The limit of a root is the root of the limit. \( \lim_{x \to c} \sqrt{f(x)} = \sqrt{\lim_{x \to c} f(x)} \), if \( f(x) \geq 0 \).

These properties ensure that we can combine limits effectively and determine continuity for complex functions.

Elementary Functions

Elementary functions play a significant role in understanding continuity. These are basic functions such as polynomials, exponentials, logarithms, and trigonometric functions. They are continuous on their domains. A key property of elementary functions is that they naturally maintain continuity.

• Polynomials: Functions with a form like \( ax^n + bx^{n-1} + \cdots + c \). Continuous everywhere on \( \mathbb{R} \).
• Rational Functions: Fractions of polynomials like \( \frac{p(x)}{q(x)} \). Continuous wherever the denominator \( q(x) eq 0 \).
• Exponential and Logarithms: The exponential function \( e^x \) is continuous everywhere, while \( \ln(x) \) is continuous for \( x > 0 \).
• Trigonometric Functions: Functions like \( \sin(x) \) and \( \cos(x) \) are continuous everywhere, except for specific domains of \( \tan(x) \), \( \csc(x) \), etc.

Understanding these functions and their properties allows us to determine continuity effortlessly within their domains. Most composite functions of elementary functions are also continuous, making them integral in calculus studies.