Problem 10 - Calculus

Question

In Exercises \(3-10\), determine the end behavior of each function as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). $$f(x)=e^{x}$$

Step-by-Step Solution

Answer

\(f(x)\) approaches infinity as \(x\to \infty\) and zero as \(x\to -\infty\).

Step 1: Identify the Function and Scenario

We are given the exponential function \(f(x)=e^x\). We need to determine its end behavior as \(x\) approaches positive infinity and negative infinity.

Step 2: Evaluate the End Behavior as \(x\rightarrow+ ext{∞}\)

As \(x\) approaches positive infinity, the function \(f(x)=e^x\) also increases towards infinity because the exponential function grows rapidly as \(x\) becomes larger.

Step 3: Evaluate the End Behavior as \(x\rightarrow- ext{∞}\)

As \(x\) approaches negative infinity, \(f(x)=e^x\) approaches zero. This is because \(e^x\) represents \(\frac{1}{e^{-x}}\) which goes to zero as \(x\) becomes negatively larger.

Key Concepts

End Behavior

Understanding the end behavior of functions is crucial to determining how a function behaves as its input values become increasingly large, either positively or negatively. For exponential functions like \( f(x) = e^x \), the end behavior describes the direction in which the graph of the function heads:

• As \( x \rightarrow +\infty \), \( e^x \) grows without bound. This means that the function values increase rapidly and tend towards infinity. The graph shoots upwards steeply as \( x \) increases.
• Conversely, as \( x \rightarrow -\infty \), \( e^x \) approaches zero. This is due to the fact that exponential decline means the function flattens out and gets closer to the x-axis without ever actually touching or crossing it.

In summary, the end behavior provides a clear picture of the long-term tendencies of the function's graph, helping us understand its most extreme tendencies.

Infinity Limits

Infinity limits focus on what happens to the function as it stretches beyond the finite boundaries, looking at its behavior towards infinity or negative infinity.For \( f(x) = e^x \):- The limit as \( x \to +\infty \) for an exponential function \( e^x \) is \( \infty \). This indicates that as \( x \) increases without bound, the function value does the same.- Conversely, the limit as \( x \to -\infty \) for \( e^x \) is \( 0 \). As \( x \) decreases into negative infinity, the exponential function rapidly diminishes and approaches zero.Infinity limits give us useful insights into how functions behave in extreme conditions, shedding light on values that might not be immediately visible in standard evaluations, but are extremely important in understanding the full scope of the function's growth or decay.

Graphical Behavior

Exponential functions like \( f(x) = e^x \) display a distinctive graphical behavior characterized by rapid growth, and sometimes, decay. Here’s a breakdown:- **Rising Rapidly:** On the graph, for positive \( x \) values, \( e^x \) rises steeply, showing sharp ascent from left to right. This illustrates exponential growth.- **Approaching Zero:** For negative \( x \) values, \( e^x \) lies very close to the x-axis. The curve shows a horizontal approach towards the axis but never truly touches it, indicating infinite decrease but never reaching zero.- **Asymptotic Nature:** The graph has an asymptote, which in this case is the x-axis, meaning the line that the curve gets infinitely closer to but never meets. The asymptotic behavior of \( f(x) = e^ x \) near negative infinity is a hallmark of exponential functions.This graphical behavior is paramount in visualizing how exponential functions function under various conditions, highlighting their unique exponential growth and approach towards zero across the axis spectrum.