Problem 20 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{t \to \infty}\frac{t - t\sqrt{t}}{2t^{3/2} + 3t - 5} \)

Step-by-Step Solution

Answer

The limit is \( \frac{-1}{2} \).

Step 1: Identify Dominant Terms

As we are dealing with limits at infinity, it's important to identify the most dominant terms in both the numerator and the denominator. Looking at the numerator \( t - t\sqrt{t} \), the dominant term is \( -t\sqrt{t} \), which simplifies to \( -t^{3/2} \). In the denominator \( 2t^{3/2} + 3t - 5 \), the dominant term is \( 2t^{3/2} \).

Step 2: Divide by the Dominant Term

To simplify the expression, divide both the numerator and the denominator by the dominant term, \( t^{3/2} \). This gives: \[ \frac{-t^{3/2}/t^{3/2} + t/t^{3/2}}{2t^{3/2}/t^{3/2} + 3t/t^{3/2} - 5/t^{3/2}} = \frac{-1 + 1/\sqrt{t}}{2 + 3/\sqrt{t} - 5/t^{3/2}} \].

Step 3: Evaluate the Limit as \( t \to \infty \)

As \( t \to \infty \), the terms \( 1/\sqrt{t} \) and \( 5/t^{3/2} \) approach zero. Thus, the simplified expression becomes: \[ \lim_{t \to \infty}\frac{-1 + 0}{2 + 0 - 0} = \frac{-1}{2} \].

Key Concepts

Dominant Terms

When dealing with calculus limits, particularly limits at infinity, identifying dominant terms is crucial. Dominant terms are essentially the "driving forces" of an expression as the variable approaches infinity.

• Look for the term with the highest power in both the numerator and the denominator.
• Higher powers of a variable dominate lower powers because they grow faster as the variable increases.

In the given exercise, for the numerator, the terms are \( t \) and \(-t\sqrt{t} \). As \( t \to \infty \), the term \(-t\sqrt{t}\) is more dominant since it can be rewritten as \(-t^{3/2}\), which is a higher power than \( t^1 \).
For the denominator, amongst \( 2t^{3/2} \), \( 3t \), and \(-5 \), the term \( 2t^{3/2} \) is dominant because \( t^{3/2} \) is the highest degree term.
By pinpointing these dominant terms, we simplify complex expressions, making it easier to find the limits.

Limits at Infinity

Limits at infinity deal with how a function behaves as the variable approaches extremely large positive or negative values. The core idea is to observe the trend of the function instead of calculating an exact number.

• For rational functions like \( \frac{polynomial}{polynomial} \), identify which polynomial terms dictate the function's behavior as \( x \to \infty \).
• Evaluate the expression using these dominant terms to understand the overall limit tendency.

In our exercise, the function's limit at infinity, \( \lim_{t \to \infty} \), is found by examining expression trends in both the numerator and the denominator after identifying dominant terms. This process often involves simplifying using division by the dominant term.
As a result, non-dominant terms become negligible and minor variations between dominant terms dictate the limit.

Simplifying Expressions

Simplifying expressions is an essential step in evaluating limits. This process involves making the expression easier to work with by focusing on dominant terms. Here are some key points:

• Divide all terms by the highest power found in the dominant terms. This reduces complicated fractions to simpler forms.
• Check how each term behaves as the variable tends to infinity. Often, the smaller terms vanish or become minimal.

For the problem at hand, dividing by \( t^{3/2} \) transforms the initial expression to a simpler one:\[ \frac{-1 + 1/\sqrt{t}}{2 + 3/\sqrt{t} - 5/t^{3/2}} \].
In this form, terms with \( 1/\sqrt{t} \) and \(-5/t^{3/2} \) proceed toward zero as \( t \to \infty \), revealing the limit of the function as \( \frac{-1}{2} \).
Such simplification is a powerful method for determining limits efficiently.