Textbook Question
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim h →0 ((x + h)² ― x²)/h
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1. Limits and Continuity
Finding Limits Algebraically
1:30 minutesProblem 2.2.13
Textbook Question
Calculating Limits
Find the limits in Exercises 11–22.
limt→6 8(t−5)(t−7)
Verified step by step guidance1
Step 1: Understand the problem. We need to find the limit of the function 8(t−5)(t−7) as t approaches 6.
Step 2: Substitute t = 6 directly into the function to check if the limit can be found by direct substitution. This is often the first approach when dealing with polynomial functions.
Step 3: Evaluate the expression 8(t−5)(t−7) at t = 6. Substitute t = 6 into the expression: 8(6−5)(6−7).
Step 4: Simplify the expression. Calculate each part: (6−5) and (6−7), then multiply these results by 8.
Step 5: If direct substitution results in a defined value, that is the limit. If it results in an indeterminate form, consider other techniques such as factoring, rationalizing, or using L'Hôpital's Rule.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. In this case, we are interested in the limit of the function as t approaches 6.
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Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, 8(t−5)(t−7), is a polynomial, and limits involving polynomials can often be evaluated by direct substitution, provided the function is continuous at the point of interest.
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Substitution Method
The substitution method is a technique used to evaluate limits by directly substituting the value that the variable approaches into the function. If the function is continuous at that point, this method yields the limit. In this exercise, substituting t = 6 into the polynomial will help find the limit as t approaches 6.
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