Problem 12 - Precalculus: Mathematical 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

-2

Step 1: Simplify the Expression

The expression given is \( \frac{1}{t} - \frac{2t}{t-1} \). Notice that both terms involve \( t \). We need to bring these terms to a common denominator to simplify the expression. The common denominator is \( t(t-1) \).

Step 2: Rewrite Each Fraction

Rewriting \( \frac{1}{t} \) with the common denominator, we multiply the numerator and the denominator by \( t-1 \), resulting in \( \frac{t-1}{t(t-1)} \). For \( \frac{2t}{t-1} \), we multiply the numerator and the denominator by \( t \), transforming it into \( \frac{2t^2}{t(t-1)} \).

Step 3: Combine the Fractions

Now, combine the fractions: \( \frac{t-1}{t(t-1)} - \frac{2t^2}{t(t-1)} = \frac{t-1 - 2t^2}{t(t-1)} \).

Step 4: Simplify the Numerator

Simplify the numerator by combining like terms: \( t - 1 - 2t^2 = -2t^2 + t - 1 \).

Step 5: Analyze the Limits

The final expression is \( \frac{-2t^2 + t - 1}{t(t-1)} \). As \( t \rightarrow \infty \), the terms in the polynomial become dominated by the highest power of \( t \), which is \( t^2 \) in both the numerator and denominator. Therefore, the expression approaches \( \frac{-2t^2}{t^2} \).

Step 6: Simplifying for Final Limit

Calculate the limit of \( \frac{-2t^2}{t^2} \). As \( t \) cancels out, this becomes \( -2 \).

Step 7: Conclusion: Determine the Final Limit

The limit of \( \lim_{t \to \infty} \left( \frac{1}{t} - \frac{2t}{t-1} \right) = -2 \).

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions as they approach a certain point or value. Here, we are interested in what happens to the value of a function as the variable gets extremely large, in this exercise it's as \( t \) approaches infinity.

When solving limits, we're looking for the output a function approaches as the input approaches a target value. This can be a number, infinity, or some undefined outcome. By using concepts of limits, we can deal with indeterminate forms and compute behaviors of functions at points of discontinuity.

• **Approaching Infinity:** Limits evaluate growth trends. If a function doesn't settle on a particular value, it might approach infinity (or negative infinity).
• **Dominating Terms:** As inputs grow large, higher power terms often dictate the function's trend. In this exercise, we see how \( t^2 \) in both the numerator and denominator becomes significant as \( t \) grows.
• **Simplifications:** Oftentimes, simplifying expressions helps in seeing trends and solving limits. By breaking down complex expressions into simpler parts, we can better understand how variables influence results.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. This context makes it important to know how they simplify and behave as variables approach certain values.

Simplifying rational expressions involves factoring and finding common denominators, similar to basic arithmetic fractions. This step is crucial, especially when evaluating limits involving expressions that can become complex.

• **Common Denominator Method:** To subtract rational expressions, finding a common denominator is fundamental. In this exercise, \( t(t-1) \) acts as the common denominator for simplifying the expressions.
• **Simplification Aids Calculation:** Simplifying the expression helps in identifying the impact of each term in regards to limit calculations. It allows clearer insight into which terms cancel out and which impact the resulting value.
• **Numerator and Denominator Relationship:** Understanding how the numerator and denominator of a rational expression interact helps in predicting the quotient's behavior as variables change.

Infinite Limits

Infinite limits describe the behavior of a function as it heads to or beyond infinity. Here, we're tasked with understanding the changes in functional value as \( t \) increases to infinity.

As \( t \) approaches infinity in a rational expression like \( \frac{1}{t} - \frac{2t}{t-1} \), the dominant terms become crucial. The function often trends towards a certain value, indicating how the expression stabilizes.

• **Dominant Terms:** By focusing on the highest power terms, we see that lower power terms become negligible. The term \( -2t^2 \) prominently inffluences the results when divided by \( t^2 \).
• **Rational Limit Outcomes:** When evaluating rational limits as variables approach infinity, terms with the highest power effectively reduce the limit to a simpler form, like in \( \frac{-2t^2}{t^2} \).
• **Understanding Results:** What's left after simplification guides us to the final result of the limit. For this exercise, \(-2\) is the result, indicating the expression stabilizes towards a negative value of 2.