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Magazine Chapter 2: Problem 54
Determine whether \(f\) is continuous at \(a\). $$ f(x)=\sqrt{x}\left(x^{2}+4\right) ; a=4 $$
Short Answer
Expert verified
The function \(f(x)\) is continuous at \(x = 4\).
Step by step solution
01
Definition of Continuity
A function \(f\) is continuous at a point \(a\) if the following three conditions are met: 1. \(f(a)\) is defined.2. The limit of \(f(x)\) as \(x\) approaches \(a\) exists.3. \(\lim_{x \to a} f(x) = f(a)\).
02
Check if \(f(a)\) is Defined
To determine if \(f(a)\) is defined, substitute \(x = 4\) into \(f(x)\):\(f(4) = \sqrt{4}(4^2 + 4) = 2(16 + 4) = 2 \times 20 = 40\).Since \(f(4) = 40\), \(f(a)\) is defined for \(a = 4\).
03
Find the Limit as \(x\) Approaches 4
Find \(\lim_{x \to 4} f(x)\):\[\lim_{x \to 4} \sqrt{x}(x^2 + 4) = \lim_{x \to 4} \sqrt{x} \times (x^2 + 4)\]Substitute \(x = 4\):\[\lim_{x \to 4} \sqrt{x} \times (x^2 + 4) = \sqrt{4} \times (4^2 + 4) = 2 \times (16 + 4) = 40\]Thus, the limit exists and is 40.
04
Compare Limit and Function Value at \(a\)
Compare \(\lim_{x \to 4} f(x) = 40\) and \(f(4) = 40\).Since the limit as \(x\) approaches 4 equals \(f(4)\), all conditions for continuity are satisfied.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limits
A limit is essentially a value that a function or sequence "approaches" as the input or index approaches some value. Limits help us understand the behavior of functions as their inputs get closer to a certain number. Understanding limits is crucial for grasping concepts in calculus, including the continuity of functions.
- A limit can exist even if the function is not defined at that point.
- If both the left-hand and right-hand limits as you approach a point are equal, the limit at that point exists.
Function Evaluation
Function evaluation refers to the process of finding the output of a function given an input. To evaluate a function at a certain point means to substitute the input value into the function and calculate the result.
When we evaluate a function, we are essentially checking its output characteristics and verifying our findings with theoretical expectations.
For example, evaluating the function \(f(x)\) at \(x = 4\) is direct and involves these steps:
When we evaluate a function, we are essentially checking its output characteristics and verifying our findings with theoretical expectations.
For example, evaluating the function \(f(x)\) at \(x = 4\) is direct and involves these steps:
- Substitute the value of \(x\) into the function.
- Calculate the expressions involved, such as square roots or powers.
- Determine the result, which helps establish whether \(f(a)\) is defined.
Continuity Conditions
A function is continuous at a point if it is defined at that point, and its limit as the input approaches the point is equal to the function's value. Let's explore these three continuity conditions:
- Existence: The function must be defined at the particular point \(a\). Without a defined function value, continuity cannot be assessed.
- Limit Existence: The limit of the function as it approaches \(a\) from either side must exist. Increasingly close inputs should converge on a single limit value.
- Equality: The actual value of the function at \(a\) must equal the calculated limit as \(x\) approaches \(a\).
Square Root Functions
Square root functions, denoted as \(f(x) = \sqrt{x}\), often appear simple, but understanding their properties is vital when assessing function behavior.
- Square root functions are only defined for non-negative values, as you can't take the square root of a negative number in the set of real numbers.
- The basic shape of a square root function is a curve that increases slowly, offering reliable predictions about function output as input values increase.
- With square root functions, it's crucial to see how they behave in composite functions, where they are multiplied, added, or involved in more complex expressions like \(f(x) = \sqrt{x}(x^2 + 4)\).
