Problem 7 - Calculus

Question

In Exercises \(4-9,\) draw a possible graph of \(f(x) .\) Assume \(f(x)\) is defined and continuous for all real \(x\). $$\lim _{x \rightarrow \infty} f(x)=-\infty \quad \text { and } \quad \lim _{x \rightarrow-\infty} f(x)=3$$

Step-by-Step Solution

Answer

The graph of \( f(x) \) approaches 3 as \( x \to -\infty \) and goes to \(-\infty\) as \( x \to \infty \).

Step 1: Understand Limits and Behavior at Infinity

We need to consider the behavior of the function \( f(x) \) as \( x \) approaches positive and negative infinity. As \( x \to \infty \), \( f(x) \to -\infty \) indicates the graph will decrease toward negative infinity. As \( x \to -\infty \), \( f(x) \to 3 \) implies the graph will approach the horizontal line \( y = 3 \).

Step 2: Analyze the Giving Conditions

The function \( f(x) \) is continuous and defined for all real numbers. This means there are no breaks, jumps, or asymptotes in the graph anywhere along the \( x \)-axis. Continuity ensures a smooth transition between extreme ends.

Step 3: Sketch the Behavior at Positive Infinity

From \( \lim_{x \to \infty} f(x) = -\infty \), the graph starting from any point on the left must end by decreasing indefinitely as \( x \) increases. Determine a part of the function where it decreases and ends at negative infinity to show this behavior.

Step 4: Sketch the Behavior at Negative Infinity

From \( \lim_{x \to -\infty} f(x) = 3 \), the graph must approach the line \( y = 3 \) as \( x \) goes to negative infinity. The curve should get closer without actually touching the \( y = 3 \) line, representing a horizontal asymptote in the negative direction.

Step 5: Connect the Behaviors

Now connect the left end (approaching \( y=3 \) from the right) with the right end (moving down towards \(-\infty\)). This could mean a linear decrease or more possibly a curve that begins near \( y = 3 \) on the left and bends downwards indefinitely.

Key Concepts

Limits at Infinity

When dealing with limits at infinity in calculus, we are exploring the behavior of functions as their input values become very large, either positively or negatively.
For instance, a limit such as \( \lim_{x \to \infty} f(x) = -\infty \) means that as \( x \) gets larger and larger, \( f(x) \) decreases without bound. This signifies that any graph of the function will descend towards negative infinity on its right side.
On the other hand, \( \lim_{x \to -\infty} f(x) = 3 \) indicates that as \( x \) decreases (or becomes more negative), \( f(x) \) gets closer and closer to 3. This does not mean \( f(x) \) will ever reach 3, but rather it "approaches" this value.

• As \( x \to \infty \): Observe behavior towards infinity.
• As \( x \to -\infty \): Look for horizontal leveling at specific values.

Understanding limits at infinity helps determine how to sketch graphs, capturing the essential behavior across extreme inputs.

Continuous Functions

A function's continuity on the real number line implies that you can draw its graph without lifting your pencil. More formally, a function \( f(x) \) is continuous for all real \( x \) if there are no jumps, breaks, or undefined points.
This notion of continuity ensures smooth transitions as you trace a graph from left to right. In our specific exercise, the function \( f(x) \) is defined and continuous across all real numbers, assuring there are no gaps as it transitions from the described limits.

• Smooth transitions: No interruptions or jumps in the graph.
• Continuity implies predictable and complete graph over the entire domain.

Because of continuity, functions behave predictably within the boundaries of the defined limits.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as \( x \) either increases or decreases without bound.
For \( \lim_{x \to -\infty} f(x) = 3 \), we recognize that the horizontal line \( y = 3 \) acts as an asymptote that the graph incessantly approaches but never actually intersects or reaches.

• Appear as horizontal lines on a graph.
• Show potential output behavior for very large (or small) input values.

The presence of these asymptotes provides valuable insight into the end behavior of functions. By acknowledging horizontal asymptotes, one can easily sketch parts of graphs that extend infinitely, knowing exactly which value these sections tend towards.