Problem 19 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Analyze the Terms

Identify the highest power of t in both the numerator and the denominator. The numerator has terms \( \sqrt{t} \) and \( t^2 \), while the denominator has terms \( 2t \) and \( -t^2 \). The dominating term in the numerator is \( t^2 \) and in the denominator is \( -t^2 \).

Step 2: Simplify the Expression

Factor \( t^2 \) out of both the numerator and the denominator. Thus, the expression becomes \[ \lim_{t \to \infty} \frac{t^2(\frac{1}{t^{3/2}} + 1)}{t^2(\frac{2}{t} - 1)}. \]

Step 3: Cancel the Common Factor

Cancel the \( t^2 \) in the numerator and the denominator: \[ \lim_{t \to \infty} \frac{\frac{1}{t^{3/2}} + 1}{\frac{2}{t} - 1}. \]

Step 4: Evaluate the Limit of Each Term

As \( t \to \infty \), both \( \frac{1}{t^{3/2}} \) and \( \frac{2}{t} \) approach 0. This simplifies the expression in the limit to \( \frac{0 + 1}{0 - 1} = -1 \).

Step 5: Conclusion

Since the simplified expression in the limit is \( -1 \), the original expression also approaches \( -1 \) as \( t \to \infty \).

Key Concepts

Dominant Terms

When analyzing the limit of a function that has polynomial-like expressions, dominant terms help us identify which parts of the function will determine its behavior as the variable approaches infinity. In the given expression, the numerator

• has the terms \( \sqrt{t} \) and \( t^2 \)

where \( t^2 \) is the dominant term because it grows the fastest as \( t \to \infty \). The denominator

• has terms \( 2t \) and \( -t^2 \)

and here, \( -t^2 \) is the dominant term. Identifying these terms simplifies the process of finding limits because they "overpower" the other terms, making them negligible as \( t \) approaches infinity. Factoring these out allows us to simplify the expression into a form that is easier to handle when evaluating its limit.

Infinity Limits

Infinity limits involve determining the behavior of a function as the variable approaches infinity. This helps in understanding the end behavior of functions in calculus. In this exercise, we are evaluating \[ \lim_{t \to \infty}\frac{\sqrt{t} + t^2}{2t - t^2}. \]As \( t \) becomes very large, the dominant terms—\( t^2 \) in both the numerator and denominator—dictate how the function behaves. By focusing on these dominant terms and simplifying using limits, we discovered that terms like \( \frac{1}{t^{3/2}} \) and \( \frac{2}{t} \) reduce to zero as \( t \) approaches infinity. This gives us a simpler expression to evaluate, leading to the realization that the limit approaches \( -1 \). When evaluating infinity limits, it's essential to consider which terms vanish and which remain influential, as they effectively determine the outcome.

Rational Functions

Rational functions are quotients of polynomials and are often encountered in limits. Understanding these functions is crucial for analyzing their behavior at infinity. For instance, in \[ \frac{\sqrt{t} + t^2}{2t - t^2}, \]we see it as a rational expression with higher degree terms in both the numerator and denominator. To find limits as \( t \to \infty \), we can simplify by identifying and canceling the dominant terms as above. In rational functions, if the highest degree terms in the numerator and denominator are the same, they typically form a fraction that guides the limit.

• If the degree of the numerator is higher, the function may approach infinity or negative infinity.
• If the degree of the numerator is lower, the function usually tends to zero.
• When the degrees are equal, the limit generally becomes the ratio of the leading coefficients.

This systematic approach helps in predicting and computing the limits of rational functions accurately, as seen in our example with the limit resulting in \( -1 \).