Problem 23 - Calculus for Biology and Medicine

Question

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{e^{x}}{1+x} $$

Step-by-Step Solution

Answer

The limit is 0 as \(x\rightarrow -\infty\).

Step 1: Identify the type of limit

We are evaluating the limit as \(x\) approaches negative infinity of the function \(\frac{e^x}{1+x}\). As \(x\) goes to negative infinity, both the numerator \(e^x\) and the denominator \(1+x\) behave differently. We'll analyze these behaviors to determine the outcome.

Step 2: Analyze the numerator

The numerator is \(e^x\). As \(x\) approaches \(-\infty\), \(e^x\) approaches \(0\) because the exponential function rapidly decreases towards zero.

Step 3: Analyze the denominator

The denominator is \(1+x\). As \(x\) approaches \(-\infty\), the value of \(1+x\) becomes dominated by \(x\), thus \(1+x\) also approaches \(-\infty\).

Step 4: Evaluate the limit using the behaviors

With the numerator approaching \(0\) and the denominator approaching \(-\infty\), the overall fraction \(\frac{e^x}{1+x}\) approaches the limit of \(0/-\infty\). Any number over \(-\infty\) approaches \(0\).

Step 5: State the final result

Thus, as \(x\) approaches \(-\infty\), the limit is \(0\).

Key Concepts

Exponential Functions

The exponential function, particularly the natural exponential function denoted as \(e^x\), is one of the most essential functions in calculus and mathematical analysis. It has some remarkable properties that make it unique:

• It grows extremely fast compared to polynomial functions as \(x\) approaches positive infinity.
• Conversely, as \(x\) approaches negative infinity, \(e^x\) decreases rapidly, approaching 0.
• The base \(e\) is an irrational number approximately equal to 2.71828, known as Euler's number.

When working with limits involving exponential functions, recognizing whether \(x\) tends towards positive or negative infinity is crucial, as it dramatically influences the behavior of \(e^x\). In the given problem, since \(x\) approaches \(-\infty\), \(e^x\) trends towards zero, simplifying the limit evaluation.

Limit Evaluation

Limit evaluation is a fundamental aspect of calculus, allowing us to understand the behavior of functions as they approach specific values or infinity. To evaluate a limit, it's vital to follow these steps:

• Identify the type of limit: Determine whether \(x\) is approaching a finite value, positive infinity \(+\infty\), or negative infinity \(-\infty\).
• Analyze individual components: Consider how the numerator and denominator behave separately as \(x\) approaches the limit.
• Simplify the expression: Use algebraic manipulation if necessary to make the mathematical analysis easier.

In the provided exercise, recognizing how \( e^x \) and \( 1+x \) behave independently as \(x\) heads towards \(-\infty\) is key. The numerator \(e^x\) going to 0 and the denominator \(1+x\) approaching \(-\infty\) helps deduce that the limit evaluates to 0.

Infinity Behavior

Infinity behavior in calculus examines how functions behave as their variables approach plus or minus infinity. Understanding this requires:

• Recognizing asymptotic behavior: Determine if the function approaches a specific value asymptotically.
• Assessing dominant terms: Identify terms that grow fastest as \(x\) becomes very large or very small.
• Applying appropriate techniques: Use L'Hôpital's Rule or dominant term approximation to find limits at infinity.

In this problem, as \(x\) approaches \(-\infty\), the focus is on how both the numerator and denominator behave. The exponential part \(e^x\) diminishes, while \(1+x\) is dominated by the \(-x\) term, causing it to become increasingly negative. Knowing these behaviors allows us to wisely conclude that the limit results in zero.