Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\frac{4-x}{2-\sqrt{x}} ; c=4 $$

Step-by-Step Solution

Answer

The function is not continuous at \(c=4\); the discontinuity is removable.

Step 1: Confirm Function is Defined at c

To check continuity at a point, first ensure the function is defined there. Substitute \(x=4\) into \(f(x) = \frac{4-x}{2-\sqrt{x}}\):\[f(4) = \frac{4-4}{2-\sqrt{4}} = \frac{0}{0}\]. The expression is not defined at \(x=4\) due to division by zero, indicating potential discontinuity.

Step 2: Calculate the Limit as x Approaches c

To check if the discontinuity can be removed, evaluate the limit\(\lim_{x \to 4} \frac{4-x}{2-\sqrt{x}}\). Recognize the \(0/0\) form and apply multiplication by the conjugate \(2+\sqrt{x}\):\[\lim_{x \to 4} \frac{(4-x)(2+\sqrt{x})}{(2-\sqrt{x})(2+\sqrt{x})} = \lim_{x \to 4} \frac{(4-x)(2+\sqrt{x})}{4-x}\]. Simplify the expression.

Step 3: Simplify and Calculate the Limit

After cancelling \((4-x)\), the expression becomes \(\lim_{x \to 4} (2+\sqrt{x})\). Now substitute \(x=4\) to get \(2+\sqrt{4} = 2+2 = 4\). Thus, \(\lim_{x \to 4} f(x) = 4\).

Step 4: Determine Type of Discontinuity

Since the function is not defined at \(x=4\) due to a \(0/0\) form, combined with the finite limit \(\lim_{x \to 4} f(x) = 4\), the discontinuity at \(x=4\) is removable. You could redefine \(f(x)\) at \(x=4\) by setting \(f(4) = 4\) to make it continuous.

Key Concepts

Removable Discontinuity

In calculus, a removable discontinuity occurs at a certain point of a function where the function is not originally defined. However, the issue can be "fixed" by properly redefining the function at that point. This happens when the limit of the function as it approaches the point exists, but the actual function is undefined or does not match the limit.

• It is often indicated by a "hole" in the graph of the function.
• To remove this discontinuity, redefine the function so that the function's value at the point equals the limit.
• Typically results from an expression that simplifies or cancels out, such as a zero-over-zero form.

In the given exercise, the function at point \(x = 4\) was undefined due to the division by zero. However, upon calculating the limit, we found that \(\lim_{x \to 4} f(x) = 4\). By redefining the function \(f(x)\) at \(x = 4\) to equal 4, the discontinuity is "removed," resulting in a perfectly continuous function at this point.

Limits in Calculus

Limits are a fundamental concept in calculus, allowing us to understand the behavior of a function as it approaches a particular point. Limits help identify the function's continuity, employing strategies to evaluate complex forms. For example,

• The limit of a function \(f(x)\) as \(x\) approaches a point \(c\) is represented as \(\lim_{x \to c} f(x)\).
• Calculating limits can help to determine if a point on the function is continuous or has a discontinuity.
• Common techniques include direct substitution, factoring, or multiplying by the conjugate to simplify indeterminate forms like \(0/0\).

In the example problem, the original expression \(\frac{4-x}{2-\sqrt{x}}\) results in a \(0/0\) form at \(x=4\), which is indeterminate. By multiplying by the conjugate \(2+\sqrt{x}\), the expression simplifies, allowing us to find the limit as 4. This result is crucial to define continuity and distinguishes between removable and non-removable discontinuities.

Function Definition

A function in mathematics is a relation that associates each element of a set with exactly one element of another set. This is often presented in algebraic form, with variables reflecting the inputs and outputs of the relation. Understanding how a function is defined helps us to explore continuity and other properties.

• A function \(f(x)\) is said to be defined at a particular \(x\) if there exists a corresponding \(y\) value.
• Improper definition or undefined points can lead to discontinuities in the function.
• To make a function continuous at a point with a removable discontinuity, one must redefine the function's value at that point to match the limit, creating a seamless graph.

In the original problem, the function \(f(x)\) is not defined at \(x=4\) due to the division by zero. However, this lack of definition indicates the potential for a removable discontinuity, which can be corrected by appropriate function definition at that point. By setting \(f(4) = 4\), we redefine the function to ensure continuity, aligning it with the calculated limit.