Problem 45 - Calculus for Biology and Medicine

Question

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right) $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Recognize the Form

First, identify if the limit is in the form that allows for the application of l'Hospital's Rule. In this problem, we need to recognize the form of \( \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right) \). As \( x \) approaches infinity, \( \frac{3}{x} \) approaches 0, so the expression inside the limit becomes \( 1 + 0 \), tending to 1.

Step 2: Determine if l'Hospital's Rule Applies

L'Hospital's Rule is typically applied to limits involving indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Our expression, when evaluating as \( x \rightarrow \infty \), is not in an indeterminate form but tends directly to 1.

Step 3: Apply the Limit Directly

Since the expression \( 1 + \frac{3}{x} \) simplifies directly to 1 as \( x \rightarrow \infty \), we don't need to use l'Hospital's Rule. We can evaluate \( \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right) = 1 + 0 = 1 \).

Key Concepts

Limits in Calculus

In calculus, studying limits is fundamental. Limits help us understand the behavior of functions as they approach certain values. This could mean as a function approaches a specific number or as it stretches towards infinity. Limits allow us to analyze situations where direct evaluation isn't feasible or possible. For example:

• Where a function reaches a certain bound.
• Where derivatives of functions are concerned.
• Where calculations of area under curves or voluminous rotations are needed.

In the context of the given exercise, the limit of the expression \( \lim_{x \rightarrow \infty}\left(1+rac{3}{x}\right) \) was examined with terms simplifying naturally. As \( x \) grows larger, the fraction \( \frac{3}{x} \) approaches zero, leading the function smoothly towards 1. This demonstrates how limits can reveal the long-term behavior of functions as they stretch towards infinite scales.

Indeterminate Forms

Indeterminate forms are unique scenarios in calculus that occur when evaluating limits directly provides results that are not expressive of true limits. These forms include:\( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( \infty - \infty \), \( 0 \times \infty \), and others. They are problematic for direct substitution because they may imply different outcomes depending on the transformation and manipulation of the expressions involved.
In this exercise, the expression given was not in an indeterminate form but directly approached 1. Recognizing this meant that l'Hospital's Rule was neither necessary nor applicable. The clear understanding of the limit marks the distinction from situations where indeterminate forms require deeper exploration or alternative tactics for limit evaluation. Thus, knowledge of indeterminate forms strengthens the strategic approach in complex calculus problems.

Infinite Limits

Infinite limits occur when the variable within a limit tends toward infinity, or alternatively, when the function's outcomes grow without boundary. They are crucial when evaluating functions that extend infinitely in domain or range. The particular scenario of an infinite limit can address questions such as:

• The behavior of asymptotes for rational functions.
• End-behavior analysis of polynomial functions.
• The exponential growth or decay in models.

In the given problem, as \( x \rightarrow \infty \), it revealed a stable horizontal trend towards a finite number, 1. While it did not require l'Hospital's Rule, understanding infinite limits is pivotal to tackling expressions with more complexity and achieving accuracy when direct evaluation isn't applicable.
Recognizing infinite limits assists in mapping graph behaviors and predicting function behavior in extreme domains, providing foundational insight into calculus studies.