Problem 7 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{t \rightarrow \infty} \frac{t}{t^{1 / 2}+2 t^{-1 / 2}} $$

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Identify the dominant term in the numerator and denominator

In the expression \( \frac{t}{t^{1/2} + 2t^{-1/2}} \), the numerator is \( t \) and the denominator is \( t^{1/2} + 2t^{-1/2} \). As \( t \to \infty \), \( t^{-1/2} \) becomes very small, so the dominant term in the denominator is \( t^{1/2} \).

Step 2: Simplify the expression by dividing by the dominant term

Divide both the numerator and the denominator by \( t^{1/2} \). This gives: \[\frac{t/t^{1/2}}{t^{1/2}/t^{1/2} + 2t^{-1/2}/t^{1/2}} = \frac{t^{1/2}}{1 + 2t^{-1}}\]

Step 3: Evaluate the limit

As \( t \to \infty \), \( t^{-1} \) approaches zero, simplifying the expression to \[\frac{t^{1/2}}{1 + 0} = t^{1/2}.\] Therefore, \( \lim _{t \rightarrow \infty} \frac{t}{t^{1 / 2}+2 t^{-1 / 2}} = \infty \).

Step 4: Conclusion

The limit of the given expression as \( t \to \infty \) is \( \infty \). This result indicates that the value of the expression increases without bound as \( t \to \infty \).

Key Concepts

Asymptotic analysis

In mathematics, asymptotic analysis is a way to describe the behavior of functions as they approach a specific value or infinity. This technique is crucial for understanding limits, especially when dealing with functions that exhibit complex behaviors as their variables tend to infinity or another boundary. In the context of the limit exercise provided, asymptotic analysis helps determine how the expression behaves as \( t \to \infty \). By focusing on the dominant terms, we can predict that

• the numerator grows rapidly as \( t \) increases, and
• the terms in the denominator \( t^{1/2} + 2t^{-1/2} \) each have their growth rates.

As \( t \to \infty \), the term \( 2t^{-1/2} \) becomes insignificant compared to \( t^{1/2} \), clearly showing the importance of considering only the most influential terms via asymptotic analysis. This results in the ability to simplify expressions for evaluating limits at infinity, leading to more straightforward calculations.

Simplifying algebraic expressions

Simplifying algebraic expressions is essential for making mathematical problems more manageable, especially when solving for limits. When we simplify an expression, we aim to reduce it to its simplest form without changing its value. In the example given, simplifying involves comparing the rates of growth of different terms.
By dividing each term by the dominant term, \( t^{1/2} \), we can see how various parts of the expression behave relative to each other. The expression \( \frac{t}{t^{1/2} + 2t^{-1/2}} \) simplifies to \( \frac{t^{1/2}}{1 + 2t^{-1}} \). This shows that as \( t \) grows larger, the term \( 2t^{-1} \) shrinks towards zero.
This method helps eliminate insignificant terms, revealing that the expression simplifies to just \( t^{1/2} \). It not only makes it easier to evaluate limits but also provides insights into the behavior of functions as variables grow large.

Dominant term identification

Identifying the dominant term in an algebraic expression is crucial when finding limits, especially as the variable approaches infinity. The dominant term dictates the behavior of the entire expression at boundary values since it grows at the fastest rate compared to the other terms in the expression.
In our example

• the numerator is \( t \), which is straightforward, and
• the denominator is \( t^{1/2} + 2t^{-1/2} \).

As \( t \to \infty \), \( t^{1/2} \) becomes the dominant term in the denominator. This is because \( 2t^{-1/2} \) becomes negligible for very large \( t \).
By isolating the dominant term, we reduce the complexity of evaluating the limit. In practical terms, this identification guides us in simplifying the expression, immediately signaling that the other terms become less impactful. Understanding which term dominates allows us to predict the long-term behavior of an algebraic expression as it approaches infinity.