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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.13
Chapter 2, Problem 2.2.13

Calculating Limits


Find the limits in Exercises 11–22.


limt→6 8(t−5)(t−7)

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1
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.

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Key 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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Introduction to Polynomial Functions

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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Substitution With an Extra Variable
Related Practice
Textbook Question

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = 3x - 4, x = 2

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Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


P(θ)=θ³ − 4θ² + 5θ; [1,2]

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Textbook Question

Theory and Examples


Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

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Textbook Question

A function value Show that the function F(x) = ( x − a)²(x − b)² + x takes on the value (a + b)² for some value of x.

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Textbook Question

Formal Definitions of One-Sided Limits


Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

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Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (x² − x + sin x) / 2x

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