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z² + 4z + 4 f(2) = - 4z +4 f(2) =미 lim f(2) = 1. Z-12

z² + 4z + 4 f(2) = - 4z +4 f(2) =미 lim f(2) = 1. Z-12

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Suppose that \( p(x) \) and \( q(x) \) are polynomials. The expression \[ \lim_{{x \to a}} \frac{{p(x)}}{{q(x)}} = L \] means that as \( x \) gets very close to \( a \)—without being \( a \), the output \(\frac{{p(x)}}{{q(x)}}\) gets very close to \( L \). How to find \( L \)? - If \( q(a) \neq 0 \), then \( L = \frac{{p(a)}}{{q(a)}} \). That's simply the evaluation of the function at \( a \). - If \( p(a) = 0 \) and \( q(a) = 0 \), then \( p(x) \) and \( q(x) \) have a common factor. Factor both polynomials and cancel the common factors out. Then \( L \) is the limit of the equivalent function. - If \( p(a) \neq 0 \) and \( q(a) = 0 \), find the one-sided limits and compare them. **Practice** 1. \( f(x) = \frac{{x^2 + 4x + 4}}{{x^2 - 4x + 4}} \) - \( f(2) = \) \(\underline{\qquad}\) - \(\lim_{{x \to 2}} f(x) = \) \(\underline{\qquad}\) 2. \( f(x) = \frac{{x^2 + 2x}}{{x^2 - 2x}} \) - \( f(2) = \) \(\underline{\qquad}\) - \(\lim_{{x \to 2}} f(x) = \) \(\underline{\qquad}\) 3. \( f(x) = \frac{{x^2 + 4x + 4}}{{x^2 - 4}} \) - \( f(2) = \) \(\underline{\qquad}\) - \(\lim_{{x \to 2}} f(x) = \) \(\underline{\qquad}\) 4. \( f(x) = \frac{{x^2 - 4x + 4}}{{x^2 - 4}} \) - \( f(
Transcribed Image Text:Suppose that \( p(x) \) and \( q(x) \) are polynomials. The expression \[ \lim_{{x \to a}} \frac{{p(x)}}{{q(x)}} = L \] means that as \( x \) gets very close to \( a \)—without being \( a \), the output \(\frac{{p(x)}}{{q(x)}}\) gets very close to \( L \). How to find \( L \)? - If \( q(a) \neq 0 \), then \( L = \frac{{p(a)}}{{q(a)}} \). That's simply the evaluation of the function at \( a \). - If \( p(a) = 0 \) and \( q(a) = 0 \), then \( p(x) \) and \( q(x) \) have a common factor. Factor both polynomials and cancel the common factors out. Then \( L \) is the limit of the equivalent function. - If \( p(a) \neq 0 \) and \( q(a) = 0 \), find the one-sided limits and compare them. **Practice** 1. \( f(x) = \frac{{x^2 + 4x + 4}}{{x^2 - 4x + 4}} \) - \( f(2) = \) \(\underline{\qquad}\) - \(\lim_{{x \to 2}} f(x) = \) \(\underline{\qquad}\) 2. \( f(x) = \frac{{x^2 + 2x}}{{x^2 - 2x}} \) - \( f(2) = \) \(\underline{\qquad}\) - \(\lim_{{x \to 2}} f(x) = \) \(\underline{\qquad}\) 3. \( f(x) = \frac{{x^2 + 4x + 4}}{{x^2 - 4}} \) - \( f(2) = \) \(\underline{\qquad}\) - \(\lim_{{x \to 2}} f(x) = \) \(\underline{\qquad}\) 4. \( f(x) = \frac{{x^2 - 4x + 4}}{{x^2 - 4}} \) - \( f(
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