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(a) lim f(x) x→-∞ (b) lim f(x) x→-2 (c) lim f(x) x→2- (d) lim f(x) x→2+

(a) lim f(x) x→-∞ (b) lim f(x) x→-2 (c) lim f(x) x→2- (d) lim f(x) x→2+

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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Question

Let ?(?) = x2/x2-4. Sketch the graph of f and evaluate each. Use appropriate superscripts 

The following notation represents different types of limits from calculus. These are essential concepts for understanding how functions behave as inputs approach specific values or infinity. Below are the specific limits depicted in the image: (a) \(\lim_{{x \to -\infty}} f(x)\): This denotes the limit of \(f(x)\) as \(x\) approaches negative infinity. It refers to the behavior of the function \(f(x)\) when \(x\) becomes very large in the negative direction. (b) \(\lim_{{x \to -2}} f(x)\): This represents the limit of \(f(x)\) as \(x\) approaches \(-2\). It is used to understand the behavior of the function \(f(x)\) near \(x = -2\). (c) \(\lim_{{x \to 2^-}} f(x)\): This is the left-hand limit of \(f(x)\) as \(x\) approaches \(2\). It indicates the behavior of the function \(f(x)\) as \(x\) approaches \(2\) from values less than \(2\). (d) \(\lim_{{x \to 2^+}} f(x)\): This refers to the right-hand limit of \(f(x)\) as \(x\) approaches \(2\). It shows the behavior of the function \(f(x)\) as \(x\) approaches \(2\) from values greater than \(2\). Together, these limits help in analyzing the continuity and behavior of functions at specific points or as they trend towards infinity. Understanding these can provide deeper insights into function behavior, which is crucial for solving higher-level mathematical problems.
Transcribed Image Text:The following notation represents different types of limits from calculus. These are essential concepts for understanding how functions behave as inputs approach specific values or infinity. Below are the specific limits depicted in the image: (a) \(\lim_{{x \to -\infty}} f(x)\): This denotes the limit of \(f(x)\) as \(x\) approaches negative infinity. It refers to the behavior of the function \(f(x)\) when \(x\) becomes very large in the negative direction. (b) \(\lim_{{x \to -2}} f(x)\): This represents the limit of \(f(x)\) as \(x\) approaches \(-2\). It is used to understand the behavior of the function \(f(x)\) near \(x = -2\). (c) \(\lim_{{x \to 2^-}} f(x)\): This is the left-hand limit of \(f(x)\) as \(x\) approaches \(2\). It indicates the behavior of the function \(f(x)\) as \(x\) approaches \(2\) from values less than \(2\). (d) \(\lim_{{x \to 2^+}} f(x)\): This refers to the right-hand limit of \(f(x)\) as \(x\) approaches \(2\). It shows the behavior of the function \(f(x)\) as \(x\) approaches \(2\) from values greater than \(2\). Together, these limits help in analyzing the continuity and behavior of functions at specific points or as they trend towards infinity. Understanding these can provide deeper insights into function behavior, which is crucial for solving higher-level mathematical problems.
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