Problem 13 - Thomas Calculus

Question

Find the limits in Exercises \(11-22\) $$\lim _{t \rightarrow 6} 8(t-5)(t-7)$$

Step-by-Step Solution

Answer

The limit is -8.

Step 1: Substitute the Limit Value into the Expression

To find the limit, first substitute the value that \( t \) approaches, which is 6, into the expression \( 8(t-5)(t-7) \). This step involves replacing \( t \) with 6 in the given expression.

Step 2: Simplify the Expression

After substituting, the expression becomes \( 8(6-5)(6-7) \). Now simplify the mathematical operations within the parentheses. Calculate \( 6 - 5 \) and \( 6 - 7 \) separately.

Step 3: Perform Multiplication

Once simplified, the expression in the parentheses is \( 8(1)(-1) \). Multiply these values together to find the result of the entire expression.

Step 4: Calculate the Final Result

After performing the multiplication, \( 8 \times 1 \times (-1) = -8 \). Thus, the limit is \(-8\).

Key Concepts

Substitution Method

The substitution method is fundamental when finding the limit of a function. It's about substituting a particular value into the expression to see what happens as the variable approaches that value. In this exercise, we need to find the limit as \( t \) approaches 6. The first step is to replace \( t \) with 6 in the expression \( 8(t-5)(t-7) \). This substitution helps us understand how the function behaves near this point. By directly replacing \( t \) with 6, the expression becomes \( 8(6-5)(6-7) \). This method works effectively when there's no division by zero or any undefined operations at the substitution point. Make sure you check these before substituting to prevent any errors in your calculations.

Simplifying Expressions

Simplifying expressions is all about breaking down a complex mathematical statement into simpler, more manageable pieces. After substituting the value of \( t \), our expression turned into \( 8(6-5)(6-7) \). The next task is to simplify what's inside the parentheses:

• Calculate \( 6 - 5 \), which equals 1.
• Calculate \( 6 - 7 \), which equals -1.

When simplifying, tackle each operation step-by-step, ensuring you handle each numerical computation accurately. Simplification makes it easier to solve the expression by reducing it to its most basic components before performing further calculations.

Multiplying Values

With the simplified expression in hand, it's time to multiply the values together. In our exercise, after simplification, we reached the expression \( 8 \times 1 \times (-1) \).Multiplying these numbers, follow these steps:

• Multiply 8 by 1 to get 8.
• Then multiply 8 by -1 to get -8.

Multiplication is straightforward: remember that multiplying by -1 changes the sign of the number. This means 8 becomes -8. Thus, the final result of the expression is \(-8\), showing that the limit of the function as \( t \) approaches 6 is \(-8\). Understanding the multiplication process aids in arriving at the correct limit value efficiently.