how to read limits symbols (explain with example)

**Reading Limit Symbols in Calculus**

In calculus, a **limit** describes the behavior of a function as the input approaches a particular value. The notation used for limits is essential for understanding how a function behaves at certain points, especially where it may not be explicitly defined.

### **Limit Notation:**

The general form for writing a limit is:

\[
\lim_{{x \to a}} f(x) = L
\]

This reads as:
- **"The limit of the function \( f(x) \) as \( x \) approaches \( a \) is \( L \)."**
- \( x \to a \) means that \( x \) is getting closer and closer to \( a \), either from the left side or the right side.
- \( f(x) \) is the function whose limit we are interested in.
- \( L \) is the value that \( f(x) \) approaches as \( x \) approaches \( a \).

### **Example 1:**
\[
\lim_{{x \to 2}} (3x + 1) = 7
\]

This is read as:
- "The limit of \( 3x + 1 \) as \( x \) approaches 2 is 7."
  
Explanation:
- As \( x \) gets closer to 2, the function \( 3x + 1 \) gets closer to 7. Substituting \( x = 2 \) into the expression \( 3(2) + 1 = 7 \), so the limit is 7.

### **Example 2:**
\[
\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1
\]

This is read as:
- "The limit of \( \frac{\sin(x)}{x} \) as \( x \) approaches 0 is 1."

Explanation:
- As \( x \) approaches 0, the value of \( \frac{\sin(x)}{x} \) gets closer and closer to 1. This is a well-known trigonometric limit.

### **One-Sided Limits:**

There are two types of one-sided limits:

1. **Left-Hand Limit** (approaching from the left):
   \[
   \lim_{{x \to a^-}} f(x)
   \]
   This means \( x \) approaches \( a \) from the left (values smaller than \( a \)).

2. **Right-Hand Limit** (approaching from the right):
   \[
   \lim_{{x \to a^+}} f(x)
   \]
   This means \( x \) approaches \( a \) from the right (values larger than \( a \)).

### **Example 3:**
\[
\lim_{{x \to 3^-}} f(x) = 5 \quad \text{and} \quad \lim_{{x \to 3^+}} f(x) = 6
\]

This is read as:
- "The left-hand limit of \( f(x) \) as \( x \) approaches 3 is 5, and the right-hand limit of \( f(x) \) as \( x \) approaches 3 is 6."

Explanation:
- As \( x \) approaches 3 from values less than 3, the function approaches 5. As \( x \) approaches 3 from values greater than 3, the function approaches 6. Since the left-hand limit and right-hand limit are different, the overall limit \( \lim_{{x \to 3}} f(x) \) does not exist.

### **Key Concepts:**
- The limit tells us what the output of a function **approaches** as the input gets close to a specific point.
- Limits can be evaluated from both directions (left and right), and if they match, the limit exists at that point.
- If the left-hand and right-hand limits differ, the overall limit does **not** exist at that point.

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Do you want more details or have any specific questions?

Here are 5 related questions to expand your understanding:
1. How do you evaluate limits that approach infinity?
2. What is the concept of a limit at a discontinuity?
3. How are limits applied in defining the derivative?
4. How can you calculate limits of more complex functions involving algebraic or trigonometric expressions?
5. How do one-sided limits help in piecewise functions?

**Tip:** When evaluating limits, always check if you can simplify the function first. Simplification makes it easier to spot the behavior of the function near the value of interest.

Learn how to interpret and read limit symbols in calculus with examples. Explore how limits describe the behavior of functions as inputs approach a specific value, using examples like \lim_{{x \to 2}} (3x + 1) = 7 and \lim_{{x \to 0}} \frac{\sin(x)}{x} = 1. Suitable for high school students in Grades 11-12.

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