Problem 3 - Thomas Calculus

Question

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$f(x)=\frac{2}{x}-3$$

Step-by-Step Solution

Answer

The limit of \( f(x) \) is \(-3\) as \( x \rightarrow \infty\) and \(-3\) as \( x \rightarrow -\infty\).

Step 1: Analyze the Function

The given function is \( f(x) = \frac{2}{x} - 3 \). This function can be broken down into two components: \( \frac{2}{x} \) and \( -3 \). Consider the behavior of each part as \( x \) approaches infinity and negative infinity.

Step 2: Determine the Limit as x Approaches Infinity

As \( x \to \infty \), \( \frac{2}{x} \to 0 \) because the numerator is constant and the denominator becomes infinitely large. Therefore, the dominant term in the function is \( -3 \), so \( \lim_{{x \to \infty}} f(x) = 0 - 3 = -3 \).

Step 3: Determine the Limit as x Approaches Negative Infinity

As \( x \to -\infty \), the term \( \frac{2}{x} \to 0 \) again. Similarly, the \(-3\) term dominates the expression. Thus, \( \lim_{{x \to -\infty}} f(x) = 0 - 3 = -3 \).

Step 4: Confirm with a Visualization

Using graphing tools, plot \( f(x) = \frac{2}{x} - 3 \). You will see that as \( x \) moves towards both infinity and negative infinity, the graph approaches the horizontal line \( y = -3 \). This confirms our limits found theoretically.

Key Concepts

Infinity

Infinity is a concept representing something that is limitless or unbounded. In the context of limits in calculus, infinity is used to describe the behavior of functions as the input grows larger and larger without bound. Understanding how functions behave as they approach infinity can help us analyze and predict long-term trends in mathematical models.

When discussing limits involving infinity, like the problem of finding the limit of the function \( f(x) = \frac{2}{x} - 3 \) as \( x \to \infty \) and \( x \to -\infty \), we're interested in what happens to the value of the function as the input becomes extremely large or extremely negative.

• As \( x \to \infty \), positive and very large, the term \( \frac{2}{x} \to 0 \). This means the function approaches \(-3\).
• Similarly, as \( x \to -\infty \), where \( x \) is very large in the negative direction, \( \frac{2}{x} \to 0 \) again, and the function also approaches \(-3\).

Using the concept of infinity helps describe functions whose asymptotic behavior influences their graphical representation and solution.

Function Behavior

The behavior of a function as \( x \) approaches infinity or negative infinity tells us about its end behavior. This behavior is crucial in understanding the nature of the function as it describes what happens to the values of \( f(x) \) as \( x \) takes on extremely large positive or negative values.

For the function \( f(x) = \frac{2}{x} - 3 \), we look at each part individually. The term \( \frac{2}{x} \) becomes very small (close to zero) as \( x \) becomes large or very negatively large. The constant term \(-3\) doesn't change. Thus, the overall function approaches \(-3\) from above:

• At large values of \( x \), the function slightly decreases from \(0\) to approach \(-3\) as \( x \to \infty \).
• For large negative \( x \), the function again approaches \(-3\). The behavior here is similar, showcasing a symmetrical end behavior about the horizontal asymptote \( y = -3 \).

Understanding function behavior at the ends helps in determining if there are horizontal asymptotes, as is the case here.

Visualization of Functions

Visualizing functions is an excellent way to comprehend their behavior over different intervals. Plotting a function like \( f(x) = \frac{2}{x} - 3 \) provides a clear view of how the function behaves as \( x \) approaches infinity and negative infinity.

By plotting the graph, you witness the trend of \( f(x) \) moving closer to the horizontal line \( y = -3 \). This visualization confirms the limits calculated mathematically. Observing these trends can be more intuitive for many students, bridging the gap between algebraic solutions and practical understanding:

• When using a graphing tool, focus on the overall shape and direction of the function as \( x \) increases or decreases without bound.
• Pay attention to the asymptotic nature in this problem, where the graph approaches \(-3\) but never actually reaches it as \( x \) goes to infinity and negative infinity.

Visualization cements the connection between theoretical results and graphical interpretation, making concepts more tangible and easier to grasp.