Problem 10 - Calculus Early Transcendentals

Question

\(10-12\) Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \(f(x)=x^{2}+\sqrt{7-x}, \quad a=4\)

Step-by-Step Solution

Answer

The function is continuous at \( a = 4 \) since \( \lim_{x \to 4} f(x) = f(4) = 16 + \sqrt{3} \).

Step 1: Definition of Continuity

A function \( f(x) \) is continuous at a number \( a \) if the following condition holds: \( \lim_{x \to a} f(x) = f(a) \). This means the left-hand limit, right-hand limit, and the value of the function at \( a \) must all be equal.

Step 2: Calculate \( f(a) \)

First, we need to find the value of the function at \( a = 4 \). Substituting \( a = 4 \) into the function, we have \( f(4) = 4^2 + \sqrt{7-4} = 16 + \sqrt{3} \).

Step 3: Determine \( \lim_{x \to 4} f(x) \)

To use the properties of limits, split the function into two parts: \( x^2 \) and \( \sqrt{7-x} \). These functions are continuous everywhere in their domains.

Step 4: Calculate \( \lim_{x \to 4} x^2 \)

Since \( x^2 \) is a polynomial function, it is continuous, meaning \( \lim_{x \to 4} x^2 = (4)^2 = 16 \).

Step 5: Calculate \( \lim_{x \to 4} \sqrt{7-x} \)

The square root function \( \sqrt{7-x} \) is continuous everywhere it is defined. Substituting \( x = 4 \), we get \( \lim_{x \to 4} \sqrt{7-x} = \sqrt{7-4} = \sqrt{3} \).

Step 6: Combine Limits

Using the limit laws, combine the limits from Steps 4 and 5: \( \lim_{x \to 4} (x^2 + \sqrt{7-x}) = \lim_{x \to 4} x^2 + \lim_{x \to 4} \sqrt{7-x} = 16 + \sqrt{3} \).

Step 7: Compare \( \lim_{x \to 4} f(x) \) and \( f(4) \)

We have already calculated \( \lim_{x \to 4} f(x) = 16 + \sqrt{3} \) and \( f(4) = 16 + \sqrt{3} \). Since they are equal, the function is continuous at \( a = 4 \).

Key Concepts

Definition of Continuity

Understanding the definition of continuity is the first step in determining whether a function is continuous at a specific point. A function \( f(x) \) is continuous at a number \( a \) if the condition \( \lim_{x \to a} f(x) = f(a) \) holds. In simpler terms, for a function to be continuous at a point, three key elements must align:

• The left-hand limit as \( x \) approaches \( a \).
• The right-hand limit as \( x \) approaches \( a \).
• The actual value of the function at \( a \).

These elements must be equal for the function to be seamless at that specific point. This condition ensures there's no jump, hole, or break in the graph of the function, maintaining a smooth connection.

Properties of Limits

Grasping the properties of limits is essential to applying the continuity definition effectively. Limits help us understand the behavior of functions as they approach particular points. Here are some key properties:

• Sum of Limits: The limit of a sum, \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \), allows us to find the limit of two functions added together.
• Limit of a Constant: The limit of a constant, \( \lim_{x \to a} c = c \), states that if a function is a constant, its limit is simply the constant itself.
• Limit of a Polynomial: For polynomial functions like \( x^2 \), continuity means you can directly substitute the value of \( x \) to find the limit.
• Limit of a Square Root: The limit of a square root function \( \lim_{x \to a} \sqrt{u(x)} = \sqrt{\lim_{x \to a} u(x)} \) applies wherever the function is defined.

These properties allow for simplifying complex limits, enabling us to approach problems involving continuity effectively, as demonstrated in the exercise.

Continuous Functions

Continuous functions are those that present no interruptions in their domain. In mathematics, they are functions where you can draw the graph without lifting your pencil off the paper. Using the definition of continuity and properties of limits helps identify continuous functions throughout their domains. Here’s how:

• Polynomials: These are inherently continuous for all real numbers. For instance, \( x^2 \) naturally meets continuity criteria everywhere.
• Square Roots: The square root function is continuous where the radicand (the value under the root) is non-negative, such as \( \sqrt{7-x} \) being continuous for \( x \leq 7 \).

By checking if left and right limits and function values are the same for all points within their domains, we can confidently state whether a function is continuous. The function in our example mixed a polynomial with a square root, both continuous under their defined conditions, confirming there are no sudden jumps or holes at the evaluated point\( a \).