Problem 31 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{3}{1+2 e^{x}} $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) tends to negative infinity is 3.

Step 1: Understand the function

First, understand the given function \(f(x)=\frac{3}{1+2e^{x}}\). It is a simple quotient of a constant by a function consisting of a constant plus an exponential function.

Step 2: Approach of the limit

Next, substitute \(x\) with \(-\infty\) in the function and evaluate. The exponential function \(e^{x}\) tends to 0 as \(x\) approaches negative infinity. So, \(e^{-\infty}=0\). Therefore, the denominator \(1+2e^{x}\) becomes \(1+2(0)=1\).

Step 3: Evaluate the limit

Now that the denominator is clarified, the limit can be calculated. So, \(\lim_{x \rightarrow -\infty} \frac{3}{1+2e^{x}}= \frac{3}{1}=3\).

Key Concepts

Exponential Functions

Exponential functions are a special class of functions where a constant base is raised to a variable exponent. In the expression \(e^x\), \(e\) represents Euler's number, approximately 2.718, which is the base of natural logarithms. Exponential functions like this can model many real-world phenomena such as population growth, radioactive decay, and interest calculations.
As \(x\) becomes very large, particularly towards either infinity or negative infinity, exponential functions exhibit well-defined growth or decay patterns. This is why they are incredibly useful in understanding limits at infinity, as they help predict the behavior of functions in different contexts. In short, exponential functions can rise sharply or decay towards zero, impacting the limits significantly when they form parts of a function's expression.

Evaluating Functions

Evaluating functions involves substituting specific values for their variables to determine their output, which is crucial when calculating limits. In the context of limits at infinity, we are interested in looking at the outputs of these functions as the variable approaches infinity or negative infinity. This process helps us understand what value a function is approaching.
In our particular example, the function is \(f(x) = \frac{3}{1 + 2e^x}\). When evaluating this function as \(x\) approaches \(-\infty\), we substitute values that are significantly negative into the exponential component. As previously mentioned, substituting a large negative \(x\) results in \(e^x\) approaching zero, which simplifies the computation of the limit.
By understanding the behavior of each component of a function, especially how exponential components behave at different infinities, we can make logical deductions about the limit of the entire function.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as their variables approach a particular value or infinity. When we discuss limits at infinity, asymptotic behavior helps us understand whether a function is approaching a horizontal asymptote, a value the function may never reach but gets indefinitely close to.
In our exercise, as \(x\) becomes very large and negative, the function \( \frac{3}{1+2e^x} \) simplifies due to the exponential function \(e^x\) approaching zero. Here, the asymptotic behavior indicates that as \(x\) approaches \(-\infty\), the function approaches a horizontal asymptote at \(y = 3\).
Recognizing asymptotic behavior is vital in understanding limits, as it speaks to the stability of function behavior in extreme conditions, offering insights into long-term trends and expectations.