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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 2.7.29
Chapter 2, Problem 2.7.29

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→2 (x^2+3x)=10

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Step 1: Recall the precise definition of a limit. For the limit \( \lim_{x \to a} f(x) = L \) to hold, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \varepsilon \).
Step 2: Identify the function \( f(x) = x^2 + 3x \), the point \( a = 2 \), and the limit \( L = 10 \). We need to show that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 2| < \delta \), then \( |(x^2 + 3x) - 10| < \varepsilon \).
Step 3: Simplify the expression \( |(x^2 + 3x) - 10| \). This becomes \( |x^2 + 3x - 10| \). Factor or simplify this expression to find a form that will help relate \( \varepsilon \) and \( \delta \).
Step 4: Consider the expression \( |x^2 + 3x - 10| \). Rewrite it as \( |(x - 2)(x + 5)| \). We need to ensure that \( |(x - 2)(x + 5)| < \varepsilon \) whenever \( 0 < |x - 2| < \delta \).
Step 5: Establish a relationship between \( \varepsilon \) and \( \delta \). Choose \( \delta \) such that \( |x + 5| \) is bounded when \( x \) is near 2. For instance, if \( |x - 2| < 1 \), then \( 1 < x < 3 \), so \( 6 < x + 5 < 8 \). Use this bound to find a suitable \( \delta \) in terms of \( \varepsilon \).

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Key Concepts

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

Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formalism is essential for rigorously proving limits in calculus.
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Epsilon-Delta Relationship

In the context of limits, the ε (epsilon) represents how close f(x) must be to the limit L, while δ (delta) represents how close x must be to the point a. Establishing a relationship between ε and δ is crucial for demonstrating that as x gets sufficiently close to a, f(x) will be within ε of L, thus proving the limit exists.
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Polynomial Functions

Polynomial functions, such as f(x) = x^2 + 3x, are continuous everywhere on their domain. This continuity implies that limits can often be evaluated by direct substitution. Understanding the behavior of polynomial functions helps in applying the limit definition effectively, especially when proving limits at specific points.
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Related Practice
Textbook Question

Determine the following limits.


limz→4z−5(z2−10z+24)2{\(\displaystyle\]\lim\)_{z\(\to\)4}}\(\frac{z-5}{\left(z^2-10z+24\right)^2}\)

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Textbook Question

Suppose limx→a f(x)=L{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ }\)f\(\left\)(x\(\right\))=L. Prove that limx→a (c(f(x))=cL{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ (}\)c\(\left\)(f\(\left\)(x\(\right\))\(\right\))=cL, where cc is a constant.

Textbook Question

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

Textbook Question

Determine the following limits. 

lim x→−∞ (3x7 + x2) 

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L


d. If |x−a|<δ, then a−δ<x<a+δ.

Textbook Question

Determine the following limits.


lim x→π/2 1/√sin x − 1 / x + π/2