Problem 23 - Calculus: Early Transcendentals

Question

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\frac{5 x-2}{x^{2}-9 x+20} ; a=4$$

Step-by-Step Solution

Answer

Answer: No, the function is not continuous at x = 4.

Step 1: Check if the function is defined at x = 4

We need to determine whether the function is defined at x = 4. To do this, we'll substitute x = 4 into the function and make sure that the denominator is not zero. $$f(4)=\frac{5(4)-2}{(4)^{2}-9(4)+20}=\frac{18}{16-36+20}=\frac{18}{0}$$ The denominator is zero when x = 4, so the function is not defined at this point.

Step 2: Find the limit of the function as x approaches 4

Since the function is not defined at x = 4, it is not necessary to find the limit of the function as x approaches 4, as the function is already discontinuous at this point. However, for the sake of understanding, let's find the limit anyway: $$\lim_{x \to 4} \frac{5x-2}{x^{2}-9x+20} = \lim_{x \to 4} \frac{5(x-4)}{(x-4)(x-5)} $$ This limit cannot be found by direct substitution because it is in the indeterminate form. However, we can simplify the expression by canceling the common factor of (x - 4): $$\lim_{x \to 4} \frac{5}{x-5} $$ Now we can find the limit by direct substitution: $$\lim_{x \to 4} \frac{5}{x-5} = \frac{5}{4 - 5} = -5$$

Step 3: Compare the limit and the function's value at x = 4

In this case, the function is not defined at x = 4, so we cannot compare the limit and the function's value at x = 4. Because the function is not defined at x = 4, it is not continuous at this point. In conclusion, the function $$f(x)=\frac{5x-2}{x^{2}-9x+20}$$ is not continuous at x = 4.

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions as they approach specific points or infinity. The limit of a function at a particular point tells us what value the function approaches as the input gets closer to that point.

To calculate a limit, follow these steps:

• Identify the point where the limit is to be evaluated.
• Try substituting the value directly into the function.
• If direct substitution leads to an indeterminate form, simplify the expression using algebraic techniques.

Finding limits is crucial for determining continuity, as well as for calculating derivatives and integrals. In our exercise, we calculated the limit by simplifying the function when direct substitution was not possible.

Indeterminate Forms

Indeterminate forms occur when substituting a value into a function results in an undefined mathematical expression, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms require special techniques to evaluate, because they do not provide sufficient information about the function's behavior.

When faced with an indeterminate form:

• Attempt to simplify the function, often by factoring or canceling terms.
• Re-evaluate the limit after simplification.

In our solved example, the expression at \( x = 4 \) initially resulted in the indeterminate form \( \frac{18}{0} \). By factoring and simplifying the function, we were able to evaluate the limit and avoid the indeterminate form.

Rational Functions

Rational functions are formed by the ratio of two polynomials. The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

Key characteristics of rational functions:

• The function is undefined wherever the denominator is zero.
• Discontinuities can occur at these undefined points.
• Limits help us understand the behavior of rational functions near these discontinuities.

In the highlighted exercise, \( f(x)=\frac{5x-2}{x^{2}-9x+20} \) is a rational function. By checking where the denominator equals zero, we identified a point of discontinuity at \( x = 4 \). This analysis plays a vital role in discussing continuity and understanding the function's behavior.