Question

Use a graph to explain the meaning of \(\lim _{x \rightarrow a} f(x)=\infty\)

Short answer

Answer: The graph of a function with a limit approaching infinity signifies that the value of the function, f(x), increases without bound as the input x approaches a specific value, called a vertical asymptote. It indicates that the function becomes infinitely large in the positive direction as x gets closer to this value.

Step-by-step solution

Understand the concept of a limit

A limit is used to describe the behavior of a function as its input (x) approaches a specific value. In our exercise, we want to understand the meaning of \(\lim_{x \rightarrow a} f(x) = \infty\). This means that as x approaches a, the value of the function f(x) increases without bound. Basically, it means that f(x) becomes infinitely large as x gets closer and closer to a.

Prepare the graph

In order to graphically explain the meaning of \(\lim_{x \rightarrow a} f(x) = \infty\), we need to draw the x-y plane with the function f(x) on it. To do this, choose a function f(x) that has a limit of infinity as x approaches a certain value a. A common example is the function \(f(x)=\dfrac{1}{x-2}\), where x cannot be 2. Let's use this function and draw the x-y plane, including both axes and the function itself.

Analyze the graph

When you have drawn the graph of \(f(x)=\dfrac{1}{x-2}\), you should observe that there is a vertical asymptote at x=2. An asymptote is a line that the function approaches but never actually reaches. In this case, the function approaches the vertical asymptote as x approaches 2. Observe how the function behaves when x is close to 2 but not equal to 2. You will see that as x approaches 2 from either side, f(x) becomes larger and larger in the positive direction, representing infinite growth.

Explain the graph's meaning

Now that we have analyzed the graph, we can draw conclusions about the limit. In the case of \(f(x)=\dfrac{1}{x-2}\), the graph demonstrates that as x approaches 2 (from either side), the function f(x) increases without bound. Therefore, the graph visually shows the meaning of \(\lim_{x \rightarrow a} f(x) = \infty\) by approaching a vertical asymptote. The value of the function f(x) is not defined at x=2, but the behavior of the function as x approaches 2 is what the limit describes, which in this case is an infinite limit.

Key concepts

Infinite Limits

Infinite limits describe the behavior of a function as it approaches infinity. When we say \( \lim_{x \rightarrow a} f(x) = \infty \), it means as the input \( x \) gets closer to \( a \), the output \( f(x) \) increases endlessly. This concept helps us understand functions that grow without bounds in certain conditions.
For instance, imagine a function \( f(x) = \frac{1}{x - 2} \). As \( x \) approaches 2, \( f(x) \) grows larger and larger towards infinity. This happens because you are essentially dividing a smaller and smaller number by something very close to zero, causing the output to shoot upwards.

• It’s important to note that this makes the function undefined at the point \( x = 2 \) but indicates the behavior around that point.
• An infinite limit provides a sense of how the function behaves as it gets near a particular value.

Understanding this can be crucial for recognizing situations where values grow unbounded, such as in physics or finance, helping to predict or analyze trends over a domain.

Vertical Asymptotes

Vertical asymptotes are lines that a function approaches but never actually touches or crosses. They represent the infinite boundary of a graph. When a function like \( f(x) = \frac{1}{x - 2} \) has a vertical asymptote at \( x = 2 \), it indicates that the function values are traveling upwards or downwards indefinitely as \( x \) nears 2.
These lines are crucial to understanding the plotting of graphs, as they divide the graph into distinct branches.

• Vertical asymptotes occur where the function's denominator is zero and leads to undefined behavior in a rational function.
• They provide a visual marker that highlights the dramatic change (or explosion) in the function's values.

Remember, while the function never quite reaches an asymptote, it gets infinitely close to it as the variables approach certain values.
This is particularly useful to know when analyzing and sketching functions, ensuring you can interpret real-world scenarios that involve rapid changes.

Limit of a Function

The limit of a function helps describe the function's behavior as the input approaches a certain value. It can show how a function behaves even if it is not actually reaching a particular point. This is especially useful when a function becomes undefined at a given point.
With limits, such as \( \lim_{x \rightarrow a} f(x) = \infty \), you understand the behavior without needing the function to "touch" that value.

• Limits are essential when dealing with discontinuities in functions.
• They provide insights into function trends and behaviors near specific points, even when direct substitution is not possible.

The limit helps us model real-life scenarios where variables change continuously, offering predictions and explanations for future and unknown behaviors.
Grasping the concept of limits allows one to tackle complex calculus problems, aiding in fields ranging from engineering to economics.