Problem 27 - Thomas Calculus

Question

Limits of quotients Find the limits in Exercises \(23-42\) $$\lim _{t \rightarrow 1} \frac{t^{2}+t-2}{t^{2}-1}$$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{2} \).

Step 1: Identify the Limit Expression

We need to evaluate the limit of the function as \( t \) approaches 1 for the expression \( \frac{t^{2}+t-2}{t^{2}-1} \). This involves finding the behavior of the function near \( t = 1 \).

Step 2: Attempt Direct Substitution

Substitute \( t = 1 \) directly into the function: \( \frac{1^{2}+1-2}{1^{2}-1} = \frac{0}{0} \). This results in an indeterminate form, which implies that further simplification is needed.

Step 3: Factor the Numerator and the Denominator

Factor the numerator \( t^{2} + t - 2 \) as \((t - 1)(t + 2)\), and the denominator \( t^{2} - 1 \) as \((t - 1)(t + 1)\). The expression now becomes \( \frac{(t - 1)(t + 2)}{(t - 1)(t + 1)} \).

Step 4: Cancel Common Factors

Since \( t eq 1 \) near the limit, we can cancel the common factor \( (t - 1) \) from the numerator and the denominator. The expression simplifies to \( \frac{t + 2}{t + 1} \).

Step 5: Evaluate the Limit with Simplified Expression

Substitute \( t = 1 \) into the simplified function: \( \frac{1 + 2}{1 + 1} = \frac{3}{2} \). Thus, the limit is \( \frac{3}{2} \).

Key Concepts

Indeterminate Forms

When solving limit problems, you may come across expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These are known as indeterminate forms. When you first substitute the value of \( t \) into a function, if the result is an indeterminate form, it means the expression doesn't give a straightforward result.

• Indeterminate forms indicate that more work is needed to solve the limit.
• They occur because the limit doesn't simply equal substitution value.

In our example, substituting \( t = 1 \) into the expression \( \frac{t^{2}+t-2}{t^{2}-1} \) results in \( \frac{0}{0} \). This tells us to use algebraic techniques to further manipulate the equation.
Identifying indeterminate forms helps guide us to apply other methods for finding the limit.

Factoring Expressions

Factoring expressions is a key step in solving limits when you face indeterminate forms. The technique involves rewriting a polynomial as a product of its factors. In this context, factoring helps in simplifying the given expression.
For our problem:

• The numerator \( t^2 + t - 2 \) can be factored as \((t - 1)(t + 2)\).
• The denominator \( t^2 - 1 \) is a difference of squares, which we factor as \((t - 1)(t + 1)\).

Factoring allows us to see the common terms that can be canceled out. This process often converts an indeterminate form into a determinable expression. Notably, factoring reduces complexity and helps find the legitimate value of the limit.

Canceling Terms

After factoring, the expression \( \frac{(t - 1)(t + 2)}{(t - 1)(t + 1)} \) shows the common factor \( (t - 1) \) in the numerator and denominator. Cancelling this mutual term simplifies the expression, making it easier to evaluate the limit.

• Removing common factors helps eliminate the source of the indeterminate form.
• In this approach, valid only under the condition where \( t eq 1 \), since division by zero is undefined.

Canceling these terms gives us the simpler expression \( \frac{t + 2}{t + 1} \).
This newly simplified form allows direct substitution of \( t = 1 \) without encountering indeterminate forms again, making it possible to evaluate the limit as \( \frac{3}{2} \). This process emphasizes the power of simplifying through cancellation.