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Chapter 2: Problem 5

Find the limits in Exercises \(1-18\) $$ \lim _{t \rightarrow 6} 8(t-5)(t-7) $$

Short Answer

Expert verified
The limit is -8.

Step by step solution

01

Substitute the Variable

To find the limit of the expression \( 8(t-5)(t-7) \) as \( t \) approaches 6, we begin by substituting \( t = 6 \) into the expression. This means we need to calculate \( 8(6-5)(6-7) \).
02

Simplify Each Term

First, calculate \( 6-5 \), which equals 1. Then, calculate \( 6-7 \), which equals -1. This simplifies the expression to \( 8(1)(-1) \).
03

Multiply to Find the Value

Next, perform the multiplication: \( 8 \times 1 \times -1 = -8 \). Thus, the expression simplifies to \(-8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution in Limits
When dealing with limits, substitution is a straightforward yet vital technique. It involves directly replacing the variable in the expression with the number it approaches. In our exercise, we want to find the limit of the expression as the variable \( t \) approaches 6. We substitute \( t = 6 \) into the expression \( 8(t-5)(t-7) \).
This step transforms the expression into \( 8(6-5)(6-7) \). Here are some general tips for substitution in limits:
  • Always ensure direct substitution won't result in an undefined expression, such as division by zero.
  • If direct substitution makes the expression undefined, consider using techniques like factoring, multiplying by a conjugate, or using L'H么pital's Rule if applicable.
Simplifying Expressions
Simplifying expressions is crucial for solving limits efficiently. It involves breaking down an expression into simpler terms. In the current example, after substituting \( t = 6 \), we have the expression \( 8(6-5)(6-7) \).
The next step is to simplify each component:
  • First, calculate the value of \( 6-5 \), which results in 1.
  • Then, solve \( 6-7 \), which gives us -1.
Now, the expression becomes \( 8 imes 1 imes -1 \). Simplifying expressions helps:
  • Prevent errors in calculation.
  • Reveal opportunities to factor or cancel terms in more complex problems.
Calculating Limits
The final step in handling limits is performing the necessary calculations to find the limit's value. After substitution and simplification, our task becomes straightforward arithmetic. With the expression \( 8 imes 1 imes -1 \), we multiply the numbers together:
  • Start with \( 8 imes 1 \), resulting in 8.
  • Then, multiply 8 by -1, which yields -8.
Thus, the value of the limit is \(-8\).
Calculating limits requires careful arithmetic, especially in more complicated expressions or when using methods like dividing limits or applying L'H么pital's Rule. Finally, recalculating to verify your work ensures your results are reliable and accurate.

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Most popular questions from this chapter

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