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Limits

Sarah Miller

Sarah Miller

7 min read

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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
  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 kk is a constant, then: limxak=k\lim_{{x \to a}} k = k

Limit of a Multiple of a Function

If kk is a constant and limxaf(x)=L\lim_{{x \to a}} f(x) = L, then: limxa(kf(x))=kL\lim_{{x \to a}} (k f(x)) = kL

Limit of a Sum or Difference of Functions

If limxaf(x)=L\lim_{{x \to a}} f(x) = L and limxag(x)=M\lim_{{x \to a}} g(x) = M, then: limxa(f(x)±g(x))=L±M\lim_{{x \to a}} (f(x) \pm g(x)) = L \pm M

Limit of a Product of Functions

If limxaf(x)=L\lim_{{x \to a}} f(x) = L and limxag(x)=M\lim_{{x \to a}} g(x) = M, then: limxa(f(x)g(x))=LM\lim_{{x \to a}} (f(x) \cdot g(x)) = L \cdot M

Limit of a Quotient of Functions

If limxaf(x)=L\lim_{{x \to a}} f(x) = L and limxag(x)=M\lim_{{x \to a}} g(x) = M with M0M \ne 0, then: limxaf(x)g(x)=LM\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{L}{M}

Limit of the Power of a Function

If limxaf(x)=L\lim_{{x \to a}} f(x) = L and nn is a real number, then: $...

Question 1 of 12

What is the value of limx57\lim_{{x \to 5}} 7? 🎉

0

5

7

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