Problem 42 - 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. $$ f(x)=\frac{x^{2}-100}{x-10} ; c=10 $$

Step-by-Step Solution

Answer

The function has a removable discontinuity at \( x = 10 \).

Step 1: Check the Condition for Continuity at c

A function is continuous at a point if the limit of the function as it approaches the point equals the value of the function at that point. Therefore, first plug in \( c = 10 \) into \( f(x) \). However, \( f(x) \) is \( f(x) = \frac{0}{0} \) when \( x=10 \), which is undefined. So we need to check the limit to see if it exists.

Step 2: Compute the Limit as x Approaches c

To find the limit as \( x \to 10 \), simplify \( f(x) \). Notice the numerator can be factored: \[ x^2 - 100 = (x-10)(x+10) \] Now, substitute back into \( f(x) \): \[ f(x) = \frac{(x-10)(x+10)}{x-10} \]Cancel the common \( (x-10) \) terms: \[ f(x) = x + 10 \text{ for } x eq 10 \]Now, calculate the limit: \[ \lim_{{x \to 10}} f(x) = \lim_{{x \to 10}} (x + 10) = 10 + 10 = 20\]

Step 3: Determine Function Value at Point c

Since \( f(10) \) is undefined (\( \frac{0}{0} \)), and the limit \( \lim_{{x \to 10}} f(x) = 20 \) is not equal to any defined value of \( f(10) \), the function is not continuous at \( x=10 \).

Step 4: Classify the Discontinuity

Since the limit exists and providing a value for \( f(x) \) at \( x=10 \) equals the limit would make the function continuous, the discontinuity at \( x=10 \) is removable. Thus, the function has a removable discontinuity at \( x=10 \). The function can be redefined as: \[ g(x) = \begin{cases} x+10, & x eq 10 \ 20, & x = 10 \end{cases}\]

Key Concepts

Limits

To determine the continuity of a function at a given point, we first explore limits. A limit examines the behavior of a function as its input approaches a certain value. In calculus, this is a crucial tool to assess continuity. For example, when evaluating

• \( \lim_{{x \to c}} f(x) \)

we are essentially asking, "What value is \( f(x) \) approaching as \( x \) gets arbitrarily close to \( c \)?"
Sometimes direct substitution in a function leads to indeterminate forms, like \( \frac{0}{0} \). In such cases, further simplification is necessary to find the limit. By simplifying and evaluating the limit, as demonstrated in the original problem, you can often discern how the function behaves near a point of discontinuity.

Removable Discontinuity

Discontinuity in functions represents breaking points where the function does not flow smoothly. However, not all discontinuities are alike. A removable discontinuity occurs when

• the limit of a function exists at a point,
• but the function is either undefined or does not match this limit at that point.

Such discontinuities can be "fixed" or "removed" by redefining the function value at the problematic point. To identify a removable discontinuity, check if the limit exists at the point and determine if this differs from the function's actual (or undefined) output.
For instance, in the exercise, the limit at \( x=10 \) was 20, but \( f(10) \) was undefined. By setting \( f(10) = 20 \), continuity is restored, turning the initially problematic spot into a seamless part of the function.

Factoring Polynomials

Factoring polynomials is a mathematical technique used to simplify equations or expressions. It's especially useful when dealing with rational expressions, where a common factor in the numerator and denominator can be canceled.
In the given exercise, the polynomial \( x^2 - 100 \) was factored to \( (x-10)(x+10) \). Factoring out these terms allows you to simplify the function, revealing hidden structures or removable terms. By canceling the \( (x-10) \) in both the numerator and the denominator, the function was simplified, eliminating the indeterminate form encountered at the point \( x=10 \).
This step not only aids in evaluating limits but also uncovers any removable discontinuities, enabling a clearer analysis of the function's behavior around specific points.

Function Evaluation

Function evaluation involves plugging values into a function to understand its specific outputs and potentially its continuity. When checking continuity, you substitute the point you're evaluating into the function, expecting to either find a concrete value or discover an indeterminate form that hints at a discontinuity.
In our example, substituting \( x = 10 \) into \( f(x) \) resulted in \( \frac{0}{0} \), an undefined form which cannot directly provide useful information about the function's behavior. Hence, further insights were sought through limit calculation and simplification. Proper function evaluation ensures that the behavior of a function is fully understood both at isolated points and in general.