Problem 56 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ F(x)=\frac{1}{x-3} ; \quad \text { find } \lim _{x \rightarrow 3} F(x) \text { and } \lim _{x \rightarrow 4} F(x). $$

Step-by-Step Solution

Answer

\( \lim_{x \to 3} F(x) \) does not exist; \( \lim_{x \to 4} F(x) = 1 \).

Step 1: Understanding Function Behavior

The function given is \( F(x) = \frac{1}{x-3} \). This is a rational function with a vertical asymptote at \( x = 3 \), because the denominator becomes zero, making the function undefined at this point.

Step 2: Finding the First Limit

To find \( \lim_{x \to 3} F(x) \), observe that as \( x \) approaches 3 from the left, the values of \( F(x) \) become very negative, and as \( x \) approaches 3 from the right, the values become very positive. Thus, \( \lim_{x \to 3} F(x) \) does not exist because the left and right limits do not equal each other.

Step 3: Finding the Second Limit

Next, we find \( \lim_{x \to 4} F(x) \). The function at \( x = 4 \) is \( F(4) = \frac{1}{4-3} = 1 \). Since the limit as \( x \to 4 \) is just \( 1 \), this limit is well-defined and exists.

Key Concepts

Rational Functions

Rational functions are expressions that are represented by the ratio of two polynomials. In other words, they can be written in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). These functions are significant in mathematics because they can model a wide variety of real-world situations.

Key characteristics of rational functions include:

• They are continuous except at points where the denominator equals zero, which leads to undefined values.
• The domain of a rational function consists of all real numbers except those that make the denominator zero.

Understanding rational functions involves analyzing their behavior, including identifying points of discontinuity and asymptotic behavior. Because of their algebraic form, rational functions can have asymptotes, which we will explore next.

Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the context of rational functions, vertical asymptotes occur at values of \( x \) that make the denominator of the function equal to zero, creating undefined points on the graph.

For the given function \( F(x) = \frac{1}{x-3} \), there is a vertical asymptote at \( x = 3 \). This is because the denominator becomes zero when \( x = 3 \), suggesting that \( F(x) \) is undefined there. As \( x \) approaches these points of discontinuity from either side, the output values of the function can either increase or decrease without bound, which is a common behavior found near vertical asymptotes.

While graphing, you will notice that the function curve gets infinitely closer to the line \( x = 3 \) but never actually meets or crosses it. Recognizing vertical asymptotes is crucial when analyzing the limits and behavior of rational functions.

Limit Existence

In calculus, the concept of limits helps us understand the behavior of functions as the input values approach a certain point. However, for a limit to exist at a particular point, the function must approach the same value from both the left and right sides.

When evaluating the limit \( \lim_{x \to 3} F(x) \) for the function \( F(x) = \frac{1}{x-3} \), it becomes apparent that the limit does not exist. As \( x \to 3^- \) (from the left), \( F(x) \) trends toward negative infinity, while as \( x \to 3^+ \) (from the right), \( F(x) \) trends toward positive infinity. Because these one-sided limits do not match, the overall limit at this point cannot be determined and thus does not exist.

On the other hand, for \( \lim_{x \to 4} F(x) \), the function approaches a value of \( 1 \) from both sides, confirming that this limit does exist. Examining limits thoroughly is essential for understanding the behavior of rational functions near their asymptotes and discontinuities.