Problem 28 - Precalculus: Mathematics for Calculus

Question

Estimating Limits Graphically Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{1}{1+e^{1 / x}}$$

Step-by-Step Solution

Answer

The limit does not exist because the left and right limits differ.

Step 1: Understand the expression

The exercise requires us to find the limit of the function \( f(x) = \frac{1}{1 + e^{1/x}} \) as \( x \) approaches 0. We need to check if this limit exists and estimate its value graphically.

Step 2: Graph the function

Using a graphing calculator or software, plot the function \( f(x) = \frac{1}{1 + e^{1/x}} \). Observe how the function behaves as \( x \) gets very close to 0 from both the left (negative values) and the right (positive values).

Step 3: Observe the behavior as x approaches 0 from the right

As \( x \) approaches 0 from the right side (i.e., \( x > 0 \)), the term \( e^{1/x} \) becomes increasingly larger, making the denominator \( 1 + e^{1/x} \) very large. Hence, the value of the fraction \( \frac{1}{1+e^{1/x}} \) approaches 0.

Step 4: Observe the behavior as x approaches 0 from the left

As \( x \) approaches 0 from the left (i.e., \( x < 0 \)), \( 1/x \) tends to negative infinity, leading \( e^{1/x} \) to approach 0. Thus, \( 1 + e^{1/x} \) is approximately 1, and the fraction \( \frac{1}{1+e^{1/x}} \) approaches 1.

Step 5: Determine the limit

The function approaches different values from the left and the right. From the left, it approaches 1, and from the right, it approaches 0. Hence, as \( x \rightarrow 0 \), the left and right limits differ, indicating that the limit of \( \frac{1}{1+e^{1/x}} \) as \( x \rightarrow 0 \) does not exist.

Key Concepts

Graphical Estimation

Graphical estimation is a valuable tool in calculus, particularly when analyzing limits. By plotting a graph of a function, we can visually inspect the behavior of the function as it approaches a certain point, like zero in this case.
You need to use a graphing device, such as a graphing calculator or software, to plot the function you're interested in. For this exercise, plot the function \( f(x) = \frac{1}{1 + e^{1/x}} \). As \( x \) approaches zero, observe how the graph behaves on both sides of zero.

• Is it moving towards a specific value?
• Are the left and right sides converging to the same point?

This visual method offers an intuitive understanding of whether a limit exists and allows estimation of its value to a specific degree of accuracy, often to two decimal places.

Limit Does Not Exist

A limit does not exist if the function approaches different values from either side of the point. In this problem, as \( x \) approaches 0, the values of the function differ depending on the direction.

• Approaching from the right (\( x > 0 \)), the function \( \frac{1}{1 + e^{1/x}} \) approaches 0 because the exponential term grows large.
• From the left (\( x < 0 \)), the exponential term becomes near zero, and the function approaches 1.

Since these are not the same, the limit overall does not exist at that point.
It's vital to check a function graphically and analytically to conclude if the left-hand limit equals the right-hand limit. If they don't, then the limit does not exist.

Approaching Zero

When a function is described as \( x \) "approaching zero," it implies examining the behavior of the function very close to the dot zero on the axis from both positive and negative sides. The essence of limits is looking at this approaching trend, not necessarily reaching the exact value.

• From the right: Imagine \( x \) taking values closer to zero but remaining positive. Here, observe how the function value changes.
• From the left: \( x \) now takes on values just less than zero, inching closer from the negative side. Notice if it reaches or diverges from a particular value.

Understanding "approaching zero" helps in examining changes in function behavior without merely substituting \( x \). Exploratory approaches like this are integral to the study of calculus, and particularly enriching for analyzing where limits might not exist.