Problem 16 - Calculus Early Transcendentals

Question

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. $$\begin{array}{l}{\lim _{x \rightarrow 1} f(x)=3, \quad \lim _{x \rightarrow 4^{-}} f(x)=3, \quad \lim _{x \rightarrow 4^{+}} f(x)=-3}, \\ {f(1)=1, \quad f(4)=-1}\end{array}$$

Step-by-Step Solution

Answer

The function can have discontinuities at 1 and 4 based on the limits and given values.

Step 1: Understand the Limits at Specific Points

Examine the given limits: \( \lim_{x \to 1} f(x) = 3 \) indicates that as \(x\) approaches 1, \(f(x)\) should get very close to 3. At \(x = 4\), the left-hand limit \( \lim_{x \to 4^{-}} f(x) = 3 \) indicates that as \(x\) approaches 4 from the left, \(f(x)\) should approximate 3. On the right, the limit \( \lim_{x \to 4^{+}} f(x) = -3 \) indicates that approaching from the right of 4, \(f(x)\) should approximate -3.

Step 2: Analyze the Given Function Values

Check the actual values that the function takes at specific points: \(f(1) = 1\) suggests a point at (1,1) on the graph, despite the limit approaching (1,3). Similarly, \(f(4) = -1\) indicates a point at (4,-1), contrasting with the approaching values of 3 and -3 from either side.

Step 3: Plot the Key Features on the Graph

Begin by plotting the given points on the graph: (1,1) and (4,-1). Then draw horizontal lines representing where the function approaches as a limit: as \(x\) approaches 1 from either side, draw a line towards (1,3) but have a gap at \(x=1\). At \(x=4\), approach (4,3) from the left and (4,-3) from the right with open circles to indicate where the function approaches but doesn’t touch these points.

Step 4: Connect and Sketch the Function Based on Limits and Values

Connect the plotted features smoothly. At \(x = 1\), smoothly approach y = 3 from either side but dip down to the point (1,1). At \(x = 4\), have the graph approach y = 3 from the left and y = -3 from the right while including the discontinuous point at (4,-1). The function will have breaks or jumps aligning with these features.

Key Concepts

Continuous Function

A continuous function is a special kind of function where you can draw its graph without lifting your pen. It implies that small changes in the input (x-values) lead to small changes in the output (y-values). When dealing with continuous functions, the limit of the function as it approaches a point from either direction is equal to the function's value at that point.
For example, if a function is continuous at a point like 1, then:

• The limit as x approaches 1 from the left side, the limit from the right side, and the actual value of the function at 1 should all be the same.
• If the limits and the value of the function differ, then the function is not continuous at that point.

When sketching or understanding functions, recognizing points of continuity helps predict the behavior of the graph.

Discontinuous Point

A discontinuous point in a function occurs where the function is not seamless. This means that as you approach a certain point on the graph, there is a sudden jump, hole, or gap. These points are crucial because they are spots where the function fails to connect smoothly.
In our exercise, for instance:

• At x = 1, even though \( \lim_{x \to 1} f(x) = 3 \), the function value \( f(1) = 1 \), which means there's a jump or a discontinuity between the value predicted by the limit and the actual value.
• Similarly, at x = 4, there are different behaviors approaching from the left and right, which results in a discontinuous point.

Identifying discontinuous points helps us understand where we might need to adjust or examine a function more closely.

Function Graphing

Graphing a function entails plotting points and drawing lines or curves to represent the relationship between input values and output values. When you have enough detail about limits and function values, creating an accurate graph becomes easier.
Steps to graph a function:

• Start by plotting given points, like (1,1) and (4,-1), according to the problem.
• Consider where the limits direct the function to approach, even if it doesn’t reach those heights due to discontinuities—like reaching near 3 or -3 at specific x-values.
• Use open circles for limits not matching actual values to show approach, but not reaching.

This method allows a visual representation of how the function behaves across different segments.

One-sided Limits

One-sided limits are used to explore how a function behaves as it approaches a point from one specific direction, either from the left or the right. These are vital, especially when dealing with discontinuities where the behavior differs on each side.
In our context:

• For x approaching 4 from the left \( \lim_{x \to 4^{-}} f(x) = 3 \), meaning the function nears 3 as it approaches x = 4 going from lower values.
• Conversely, from the right \( \lim_{x \to 4^{+}} f(x) = -3 \), which indicates it heads towards -3 as x increases to 4.

Using one-sided limits helps define the unique characteristics of a function around points where it might not be continuous, enhancing graph precision and understanding.