Problem 81 - Precalculus : Building Concepts and Connections

Question

Consider the function \(f(x)=2+e^{-x}.\) (a) What number does \(f(x)\) approach as \(x \rightarrow+\infty ?\) (b) How could you use the graph of this function to confirm the answer to part (a)?

Step-by-Step Solution

Answer

As \(x \rightarrow+\infty\), the function \(f(x)\) approaches the number 2. This is confirmed by the graphical representation of the function, where the function gets closer and closer to the line \(y=2\) as \(x\) tends to \(+\infty\).

Step 1: Calculate the limit of \(f(x)\) as \(x\) approaches \(+\infty\)

The limit of an exponential function as \(x\) tends to \(\infty\) or \(-\infty\) has a standard form and evaluates to a certain value depending upon the nature of the function. We need to calculate the limit of \(f(x)=2+e^{-x}\) as \(x\) tends to \(+\infty\). The given function \(f(x)\) can be split into two parts: the constant 2 and the exponential function \(e^{-x}\). The limit will be given by adding the limits of these two components. As \(x\) tends to \(+\infty\), the term \(e^{-x}\) tends to 0, since any exponential function with a negative power tends to zero as \(x\) approaches \(+\infty\). Thus, the function \(f(x)\) approaches '2' as \(x\) tends to \(+\infty\).

Step 2: Visualize through graph

The graph of the function \(f(x)=2+e^{-x}\) will always be above the line \(y=2\), as \(e^{-x}\) is always positive but decreasing as \(x\) increases. Eventually, as \(x\) tends to \(+\infty\), the contribution from the \(e^{-x}\) term becomes negligible, causing the graph to appear to asymptote to the line \(y=2\), hence confirming our result from Step 1.

Step 3: Write final answer

In conclusion, as \(x\) tends to \(+\infty\), the function \(f(x)\) approaches the number 2. This can also be confirmed graphically by observing that the graph of the function becomes increasingly close to the line \(y=2\) as \(x\) increases.

Key Concepts

Exponential Decay

Exponential decay describes a process where the amount of something reduces at a rate which is proportional to its current value. When looking at the function f(x) = 2 + e^{-x}, e^{-x} illustrates this concept of exponential decay.

For students focusing on textbooks solutions, considering how exponential decay affects the behavior of a function is vital. As the value of x grows larger, e^{-x} will dwindle toward zero. It's because the base of the exponential function is less than one (e^{-x} = 1/e^x), leading to a decrease as x increases. This characteristic causes the function's output to gradually move towards the limiting value, which does not include the part of the function that is declining to zero. In this case, the function levels off at 2 because the 'decaying' part becomes insignificant as x goes to positive infinity.

Horizontal Asymptotes

Horizontal asymptotes are straight lines that a function's graph approaches but never actually reaches, no matter how far out the graph is extended horizontally. They symbolize the value that a function is getting closer to as the input either heads towards infinity or negative infinity.

For the function f(x) = 2 + e^{-x}, as x increases towards positive infinity, e^{-x} diminishes towards zero, making the value of the function increasingly closer to 2. Therefore, the line y = 2 is a horizontal asymptote of the function. Understanding this concept helps students anticipate the long-run behavior of functions and ties together the knowledge of how limits work for functions exhibiting exponential decay.

Graphical Analysis of Limits

Visualizing the limit of a function using its graph can solidify understanding of the behavior of the function as the input approaches a particular value. For sustaining graphical analysis of limits, one generally observes how the function behaves as it gets closer to a particular x-value or as x tends towards infinity.

In the case of the function f(x) = 2 + e^{-x}, plotting the graph would reveal that as x progresses towards positive infinity, the e^{-x} term diminishes and the graph starts to run parallel to the x-axis at y = 2. This visual inspection supports the limit calculation done analytically and provides a practical understanding that complements traditional step-by-step solutions found in textbooks.