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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 7
Chapter 2, Problem 7

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.


lim x→1 (4f(x))

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1
Identify the given limit: \( \lim_{{x \to 1}} f(x) = 8 \).
Recognize that the problem involves a constant multiple of a function: \( 4f(x) \).
Apply the Constant Multiple Law for limits, which states that \( \lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x) \), where \( c \) is a constant.
Substitute the given limit into the Constant Multiple Law: \( 4 \cdot \lim_{{x \to 1}} f(x) = 4 \cdot 8 \).
Conclude that the limit is \( 4 \cdot 8 \), but do not calculate the final result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this case, as x approaches 1, the limits of f(x), g(x), and h(x) are given, which are essential for evaluating expressions involving these functions. Understanding limits is fundamental in calculus as it lays the groundwork for continuity, derivatives, and integrals.
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Limits of Rational Functions: Denominator = 0

Limit Laws

Limit laws are a set of rules that allow us to compute limits of functions based on the limits of their components. For example, one important limit law states that the limit of a constant multiplied by a function is the constant multiplied by the limit of the function. This principle is crucial for simplifying the computation of limits, such as lim x→1 (4f(x)), which can be evaluated using the known limit of f(x).
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Constant Multiple Rule

The Constant Multiple Rule is a specific limit law that states if c is a constant and f(x) is a function, then lim x→a (c * f(x)) = c * lim x→a f(x). This rule allows us to factor out constants when calculating limits, making it easier to find the limit of expressions like 4f(x) as x approaches 1, where we can directly multiply the limit of f(x) by 4.
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Related Practice
Textbook Question

Use the graph of ff in the figure to find the following values or state that they do not exist. <IMAGE>

f(0)f\(\left\)(0\(\right\))

Textbook Question

Let f(x)=x2−4x−2f\(\left\)(x\(\right\))=\(\frac{x^2-4}{x-2}\). <IMAGE>

Make a conjecture about the value of limx→2x2−4x−2{\(\displaystyle\]\lim\)_{x\(\to\)2}\(\frac{x^2-4}{x-2}\)}.

Textbook Question

Estimate lim θ→0 sin 2θ / sin θ using the graph in part (a).

Textbook Question

Determine the following limits at infinity.


lim x→−∞ x^−11

8
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Textbook Question

Let f(x)=x2−4x−2f\(\left\)(x\(\right\))=\(\frac{x^2-4}{x-2}\) . <IMAGE>

Calculate f(x)f\(\left\)(x\(\right\)) for each value of xx in the following table.

Textbook Question

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).

Evaluate lim t→2.9 f(t).