Problem 20 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{t \rightarrow \infty} \frac{t^{2}+2}{t^{3}+t^{2}-1} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Dominant Terms

To find the limit as \( t \) approaches infinity, we need to focus on the terms with the highest powers in both the numerator and the denominator. In this expression, the dominant term in the numerator is \( t^2 \), and in the denominator, it is \( t^3 \).

Step 2: Rewrite the Expression

Rewrite the original limit expression focusing on the dominant terms: \[\lim _{t \rightarrow \infty} \frac{t^2(1 + \frac{2}{t^2})}{t^3(1 + \frac{1}{t} - \frac{1}{t^3})}\]This simplifies the problem to focusing on the behavior as \( t \to \infty \).

Step 3: Simplify by Factoring Out the Dominant Terms

Factor out the highest powers of \( t \) from both the numerator and the denominator:\[\lim _{t \rightarrow \infty} \frac{1 + \frac{2}{t^2}}{t(1 + \frac{1}{t} - \frac{1}{t^3})}\]Divide both the numerator and denominator by \( t^2 \).

Step 4: Evaluate the Simplified Expression

As \( t \rightarrow \infty \), the terms \( \frac{2}{t^2} \), \( \frac{1}{t} \), and \( \frac{1}{t^3} \) tend to zero. Thus, the expression simplifies to:\[\lim _{t \rightarrow \infty} \frac{1}{t} = 0\]The main focus is on the factor of \( \frac{1}{t} \), which approaches 0.

Key Concepts

Dominant Terms

When dealing with limits, especially as a variable approaches infinity, understanding dominant terms is crucial. The dominant terms in a polynomial are those with the highest degree.
These terms grow the fastest and most significantly influence the behavior of the function as the variable becomes very large.In the exercise, to find the limit as \( t \) approaches infinity for the expression \( \frac{t^2 + 2}{t^3 + t^2 - 1} \), the dominant term in the numerator is \( t^2 \), and in the denominator, it is \( t^3 \).
Identifying these allows us to see that:

• \( t^3 \) in the denominator will grow faster than \( t^2 \).
  
• The presence of \( t^3 \) has a greater effect on the function's behavior as \( t \) increases.

Focusing on these terms helps simplify the problem and understand how the function will behave at extreme values.

Asymptotic Behavior

Understanding asymptotic behavior is like recognizing the general trend or direction of a function as its variable approaches a particular point, like infinity.
As \( t \) approaches infinity, we look at how the expression changes and what it smooths out to. In our exercise, by evaluating \( \frac{t^2(1 + \frac{2}{t^2})}{t^3(1 + \frac{1}{t} - \frac{1}{t^3})} \), we determine that the terms \( \frac{2}{t^2} \), \( \frac{1}{t} \), and \( \frac{1}{t^3} \) approach zero as \( t \to \infty \).
So, the influence of these smaller terms diminishes, revealing the dominant term \( \frac{1}{t} \).
Thus, as \( t \to \infty \), the expression effectively trends toward zero, illustrating the concept of asymptotic behavior.

Infinite Limits

Infinite limits occur when the variable in a function tends towards infinity, and we want to understand the behavior of the function at that extreme.Through simplifying using the dominant terms, we found that the original expression \( \lim _{t \rightarrow \infty} \frac{t^{2} + 2}{t^{3} + t^{2} - 1} \) simplifies to \( \lim _{t \rightarrow \infty} \frac{1}{t} \).
As \( t \to \infty \), \( \frac{1}{t} \) becomes smaller and smaller, approaching zero. This demonstrates an infinite limit, where the variable reaches infinite values, leading to the function approaching zero.

• The original complexities vanish, showing how the function acts over an infinite domain.
  
• This helps us conclude that the limit of the function at infinity is 0.

Infinite limits aid in understanding the overall trend and ultimate value as the variable stretches beyond bounds, crucial for analyzing real-world problems.