#Study Notes on Limits in Mathematics

#Table of Contents

Introduction to Limits
One-sided Limits
Two-sided Limits
Nonexistent Limits
Practice Questions
Glossary
Summary and Key Takeaways
Exam Strategies

#Introduction to Limits

#What are Limits in Mathematics?

Limits help us understand the behavior of functions as their input values approach a certain point. They are fundamental to calculus and are used to define derivatives and integrals.

Key Concept

Understanding limits is crucial because they form the foundation for many concepts in calculus.

#Key Points:

Limits describe the value that a function approaches as the input approaches a certain point.
Notation: $\lim_{{x \to c}} f(x)$ indicates the limit of the function $f(x)$ as $x$ approaches $c$ .
If the limit exists, $\lim_{{x \to c}} f(x) = R$ , where $R$ is a real number.

**Example**: For the function

f(x) = \frac{1}{x}

, as

x

approaches 0, the function becomes unbounded. Hence,

\lim\_{{x \to 0}} \frac{1}{x}

does not exist.

Exam Tip

Exam Tip: Always check the behavior of the function from both sides of the point in question.

#One-sided Limits

#What are One-sided Limits?

One-sided limits focus on the value of a function as the input approaches a certain point from one side only.

#Notation:

$\lim_{{x \to c^-}} f(x)$ : Limit from the left (or below)
$\lim_{{x \to c^+}} f(x)$ : Limit from the right (or above)

**Example**: For the function

f(x) = \begin{cases} x + 1, & \text{if } x \leq 2 \\\\ 4, & \text{if } x > 2 \end{cases}

$\lim_{{x \to 2^-}} f(x) = 3$
$\lim_{{x \to 2^+}} f(x) = 4$

Key Concept

One-sided limits are essential when dealing with piecewise functions.

Practice Question

Practice Question: Find the one-sided limits for the function $g(x) = \begin{cases} x^2, & \text{if } x < 1 \\ 2, & \text{if } x \geq 1 \end{cases}$ at $x = 1$ .

#Two-sided Limits

#What are Two-sided Limits?

Two-sided limits consider the value a function approaches as the input comes from both sides of a particular point.

#Notation:

$\lim_{{x \to c}} f(x)$ : Limit as $x$ approaches $c$ from both sides.

#Key Points:

If $\lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x)$ , then the two-sided limit exists and is equal to that value.
If the one-sided limits are not equal, the two-sided limit does not exist.

**Example**: For the function

f(x) = \begin{cases} x^2 + 1, & \text{if } x \neq 0 \\\\ 0, & \text{if } x = 0 \end{cases}

$\lim_{{x \to 0}} f(x) = 1$

Common Mistake

Common Mistake: Assuming the limit exists just because the function is defined at that point.

Practice Question

Practice Question: Determine if the two-sided limit exists for the function $h(x) = \begin{cases} 2x + 3, & \text{if } x \leq 1 \\ 5, & \text{if } x > 1 \end{cases}$ at $x = 1$ .

#Nonexistent Limits

#When does the Limit of a Function at a Point Not Exist?

For some functions, limits might not exist at certain points. This can occur in three common scenarios:

#Case 1: The Function is Unbounded Near the Point

Example: $f(x) = \frac{1}{x}$ as $x$ approaches 0. #### Case 2: The Function Oscillates Near the Point
Example: $g(x) = \cos\left(\frac{1}{x}\right)$ as $x$ approaches 0. #### Case 3: The One-sided Limits are Not Equal
Example: $h(x) = \begin{cases} 0, & \text{if } x < 0 \\ \frac{1}{2}, & \text{if } x = 0 \\ 1, & \text{if } x > 0 \end{cases}$ as $x$ approaches 0. Example: For the function $f(x) = \sin\left(\frac{1}{x^2 - 1}\right)$ , the limit does not exist as $x$ approaches -1 because the function oscillates infinitely.

Practice Question

Practice Question: Explain why $\lim_{{x \to \frac{\pi}{2}}} \tan(x)$ does not exist.

#Practice Questions

Multiple Choice: What is the two-sided limit of the function $f(x) = \begin{cases} x^2, & \text{if } x < 2 \\ 3, & \text{if } x = 2 \\ x + 1, & \text{if } x > 2 \end{cases}$ as $x$ approaches 2?

a) 4

b) 3

c) 5

d) Does not exist
Short Answer: Find the one-sided limits for the function $f(x) = \begin{cases} x - 1, & \text{if } x < 3 \\ 2x, & \text{if } x \geq 3 \end{cases}$ at $x = 3$ .

#Glossary

Limit: The value that a function approaches as the input approaches a certain point.
One-sided Limit: The limit of a function as the input approaches a point from one side (left or right).
Two-sided Limit: The limit of a function as the input approaches a point from both sides.
Unbounded: A function that increases or decreases without bound as the input approaches a certain point.
Oscillates: A function that fluctuates between values as the input approaches a certain point.

#Summary and Key Takeaways

#Summary

Limits describe the behavior of functions as inputs approach certain points.
One-sided limits consider the function's value from one side, while two-sided limits consider both sides.
A limit may not exist if the function is unbounded, oscillates, or has unequal one-sided limits.

#Key Takeaways

Understand the behavior of functions near the point of interest.
Use proper notation for limits.
Check for one-sided limits to determine two-sided limits.
Know the common scenarios where limits do not exist.

#Exam Strategies

Read Carefully: Understand the problem and identify whether it asks for a one-sided or two-sided limit.
Analyze Graphs: Use graphs to visually inspect the behavior of functions near the point of interest.
Check Both Sides: Always consider the behavior of the function from both the left and right sides.
Practice: Solve various problems to become familiar with different types of limits and functions.

By mastering these strategies and concepts, you will be well-prepared to tackle limits in your exams confidently.