1. Under which circumstances for an infinite limit could you ONLY state imf(X)= DNE and not say that the Limit is also equal to either +0 or -00.2. In your explanation, describe what must be happening for the following one-sided limits:lim f(x) andlim f(x).X + c+3. Finally, provide an example function that exhibits these properties at x = 2.

Answered: 1. Under which circumstances for an…

Calculus: Early Transcendentals

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ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

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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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1. Under which circumstances for an infinite limit could you ONLY state imf(X)= DNE and not say that the Limit is also equal to either +0 or -00.
2. In your explanation, describe what must be happening for the following one-sided limits:
lim f(x) and
lim f(x).
X + c+
3. Finally, provide an example function that exhibits these properties at x = 2.

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Step 1:Definition

We define three types of infinite limits.

Infinite limits from the left: Let $f(x)$ be a function defined at all values in an open interval of the form $(b,a)$ .

If the values of $f(x)$ increase without bound as the values of $x$ (where $x<a$ ) approach the number $a$ , then we say that the limit as $x$ approaches $a$ from the left is positive infinity and we write
$\underset{x\to a^-}{\lim}f(x)=+\infty$ .
If the values of $f(x)$ decrease without bound as the values of $x$ (where $x<a$ ) approach the number $a$ , then we say that the limit as $x$ approaches $a$ from the left is negative infinity and we write
$\underset{x\to a^-}{\lim}f(x)=−\infty$ .

Infinite limits from the right: Let $f(x)$ be a function defined at all values in an open interval of the form $(a,c)$ .

If the values of $f(x)$ increase without bound as the values of $x$ (where $x>a$ ) approach the number $a$ , then we say that the limit as $x$ approaches $a$ from the left is positive infinity and we write
$\underset{x\to a^+}{\lim}f(x)=+\infty$ .
If the values of $f(x)$ decrease without bound as the values of $x$ (where $x>a$ ) approach the number $a$ , then we say that the limit as $x$ approaches $a$ from the left is negative infinity and we write
$\underset{x\to a^+}{\lim}f(x)=−\infty$ .

Two-sided infinite limit: Let $f(x)$ be defined for all $x\ne a$ in an open interval containing $a$ .

If the values of $f(x)$ increase without bound as the values of $x$ (where $x\ne a$ ) approach the number $a$ , then we say that the limit as $x$ approaches $a$ is positive infinity and we write
$\underset{x\to a}{\lim}f(x)=+\infty$ .
If the values of $f(x)$ decrease without bound as the values of $x$ (where $x\ne a$ ) approach the number $a$ , then we say that the limit as $x$ approaches $a$ is negative infinity and we write
$\underset{x\to a}{\lim}f(x)=−\infty$ .

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