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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 7a
Chapter 2, Problem 7a

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.

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1
First, let's simplify the function \( f(x) = \frac{x^2 - 4}{x - 2} \). Notice that the numerator \( x^2 - 4 \) can be factored as a difference of squares: \( (x - 2)(x + 2) \).
Rewrite the function using the factored form: \( f(x) = \frac{(x - 2)(x + 2)}{x - 2} \).
Observe that the \( x - 2 \) terms in the numerator and the denominator can be canceled out, but only for \( x \neq 2 \) to avoid division by zero. Thus, \( f(x) = x + 2 \) for \( x \neq 2 \).
Now, for each value of \( x \) in the table, substitute \( x \) into the simplified function \( f(x) = x + 2 \) to find the corresponding value of \( f(x) \).
Remember to handle the case where \( x = 2 \) separately, as the original function is undefined at this point due to division by zero. Consider the limit as \( x \) approaches 2 if needed.

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Key Concepts

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

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function f(x) = (x^2 - 4) / (x - 2) is a rational function where the numerator is a polynomial of degree 2 and the denominator is a polynomial of degree 1. Understanding rational functions is crucial for analyzing their behavior, including identifying points of discontinuity and simplifying expressions.
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Simplifying Rational Expressions

Simplifying rational expressions involves reducing the expression to its simplest form by factoring and canceling common factors in the numerator and denominator. For the function f(x) = (x^2 - 4) / (x - 2), recognizing that the numerator can be factored as (x - 2)(x + 2) allows us to simplify the expression, which is essential for evaluating the function at specific values of x.
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Evaluating Functions

Evaluating a function means substituting a specific value for the variable and calculating the output. In this context, evaluating f(x) for various values of x requires substituting those values into the simplified form of the function. This process is fundamental in calculus as it helps in understanding the function's behavior and its graphical representation.
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Related Practice
Textbook Question

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 (f(x)−g(x))

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

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Textbook Question

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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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).