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Limits

Sarah Miller

Sarah Miller

7 min read

Next Topic - Evaluating Limits Analytically

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Study Guide Overview

This study guide covers properties of limits, including limits of constant functions, sums, differences, products, quotients, powers, and composite functions. It also explores infinite limits and how limit properties apply to them. The guide provides worked examples, practice questions, a glossary of key terms, and exam tips for applying these properties effectively.

#Properties of Limits

#Table of Contents

  1. Introduction to Limit Properties
  2. Basic Limit Properties
    • Limit of a Constant Function
    • Limit of a Multiple of a Function
    • Limit of a Sum or Difference of Functions
    • Limit of a Product of Functions
    • Limit of a Quotient of Functions
    • Limit of the Power of a Function
    • Limit of a Composite Function
  3. Infinite Limits
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways

#Introduction to Limit Properties

Understanding limit properties (also known as limit theorems) is crucial for solving complex functions algebraically. These properties allow us to break down complicated functions into simpler parts, making it easier to determine their limits.

#Basic Limit Properties

#Limit of a Constant Function

If kkk is a constant, then: lim⁡x→ak=k\lim_{{x \to a}} k = kx→alim​k=k

#Limit of a Multiple of a Function

If kkk is a constant and lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L, then: lim⁡x→a(kf(x))=kL\lim_{{x \to a}} (k f(x)) = kLx→alim​(kf(x))=kL

#Limit of a Sum or Difference of Functions

If lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L and lim⁡x→ag(x)=M\lim_{{x \to a}} g(x) = Mlimx→a​g(x)=M, then: lim⁡x→a(f(x)±g(x))=L±M\lim_{{x \to a}} (f(x) \pm g(x)) = L \pm Mx→alim​(f(x)±g(x))=L±M

#Limit of a Product of Functions

If lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L and lim⁡x→ag(x)=M\lim_{{x \to a}} g(x) = Mlimx→a​g(x)=M, then: lim⁡x→a(f(x)⋅g(x))=L⋅M\lim_{{x \to a}} (f(x) \cdot g(x)) = L \cdot Mx→alim​(f(x)⋅g(x))=L⋅M

#Limit of a Quotient of Functions

If lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L and lim⁡x→ag(x)=M\lim_{{x \to a}} g(x) = Mlimx→a​g(x)=M with M≠0M \ne 0M=0, then: lim⁡x→af(x)g(x)=LM\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{L}{M}x→alim​g(x)f(x)​=ML​

#Limit of the Power of a Function

If lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = Llimx→a​f(x)=L and nnn is a real number, then: $...

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Question 1 of 12

What is the value of lim⁡x→57\lim_{{x \to 5}} 7limx→5​7? 🎉

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5

7

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