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Magazine Chapter 2: Problem 14
\(13-14\) Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. \(g(\mathrm{x})=2 \sqrt{3-\mathrm{x}}, \quad(-\infty, 3]\)
Short Answer
Expert verified
The function \(g(x) = 2\sqrt{3-x}\) is continuous on \((-\infty, 3]\).
Step by step solution
01
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.
02
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 \(x\leq 3\).
03
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 \(g(c)\).
04
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, \(\lim_{x \to 3^-} g(x) = 0\).
05
Evaluate g(3) directly
Calculate \(g(3) = 2\sqrt{3-3} = 2\times 0 = 0\).
06
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 \(x=3\).
07
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 \((-\infty, 3]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
This can be mathematically expressed as:
- The limit \( \lim_{x \to c} f(x) = f(c) \).
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 \( f(x) \geq 0 \).
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.
