Problem 14 - Precalculus: Mathematics for Calculus

Question

Find the limit. $$\lim _{t \rightarrow \infty}\left(\frac{1}{t}-\frac{2 t}{t-1}\right)$$

Step-by-Step Solution

Answer

The limit is -2.

Step 1: Combine the Fractions

Combine the fractions into a single expression:\[\frac{1}{t} - \frac{2t}{t-1} = \frac{1(t-1) - 2t \cdot t}{t(t-1)} = \frac{t-1 - 2t^2}{t(t-1)}\]

Step 2: Simplify the Numerator

Simplify the numerator of the combined fraction:\[t-1 - 2t^2 = -2t^2 + t - 1\]

Step 3: Simplify the Fraction

Express the simplified fraction:\[\frac{-2t^2 + t - 1}{t^2 - t} = \frac{-2t^2 + t - 1}{t^2(1 - \frac{1}{t})}\]

Step 4: Evaluate the Limit as t Approaches Infinity

Evaluate the limit of each term as \( t \) approaches infinity. The term \(-2t^2\) in the numerator and \(t^2\) in the denominator dominate other terms, simplifying the fraction:\[\lim _{t \rightarrow \infty} \frac{-2t^2}{t^2} = -2\]

Step 5: Final Result

Therefore, the limit is:\[\lim _{t \rightarrow \infty}\left(\frac{1}{t}-\frac{2 t}{t-1}\right) = -2\]

Key Concepts

Infinity

When we talk about limits in calculus, infinity is a concept that refers to the behavior of a function as the input approaches a very large or very small number. In the exercise, we need to determine the limit of a function involving the variable \( t \) as it moves towards infinity. This means we're interested in understanding the trend of the expression \( \frac{1}{t} - \frac{2t}{t-1} \) as \( t \) becomes very large.

Infinity itself is not a number but a concept used to describe unbounded growth. When dealing with fractions, as \( t \) approaches infinity:

• The fraction \( \frac{1}{t} \) approaches zero because dividing one by a larger and larger number results in a smaller outcome.
• Similarly, the fraction \( \frac{2t}{t-1} \) will have its behavior analyzed as \( t \), both in the numerator and denominator, grows large.

By understanding how these terms behave as \( t \) trends towards infinity, we can simplify and analyze the function's limit behavior more effectively.

Fraction Simplification

Fraction simplification is a key technique in calculus that helps in understanding and evaluating limits. Here, our main goal is to combine and simplify fractions to make the function easier to evaluate.

In our exercise, we started with the expression \( \frac{1}{t} - \frac{2t}{t-1} \). To handle this, we need to:

• Combine the fractions into a single fraction by finding a common denominator, which is \( t(t-1) \) here.
• After combining, we simplify the fraction: the numerator becomes \( t-1 - 2t^2 \), resulting in the overall expression \( \frac{-2t^2 + t - 1}{t(t-1)} \).

Fraction simplification allows us to cancel terms, focus on dominant terms, and reduce the complexity of the expressions to get a clearer insight into their limits. By focusing on the term \( -2t^2 \), it guides our understanding at infinity, as it overshadow the behavior of other terms.

Numerator and Denominator

Understanding the roles of the numerator and denominator in a fraction is crucial for evaluating limits. Each plays a different role in controlling the behavior of the fraction as the variable approaches a particular value or infinity.

In our exercise, after simplifying, we focused on the fraction \( \frac{-2t^2 + t - 1}{t^2(1 - \frac{1}{t})} \). Here's what happens:

• The numerator \( -2t^2 + t - 1 \) must be simplified to see which term grows the fastest as \( t \) becomes large. In this case, \( -2t^2 \) dominates because it's the term with the highest power of \( t \).
• The denominator simplifies to approximately \( t^2 \) when \( t \) is very large, because \( \frac{1}{t} \) becomes negligible.

By focusing on the dominant terms, \( -2t^2 \) in the numerator and \( t^2 \) in the denominator, we can simplify the complex fraction to \( \frac{-2t^2}{t^2} \), which easily evaluates to \(-2\) as \( t \rightarrow \infty \). This technique streamlines the process of finding limits by highlighting which components impact the fraction's behavior the most.