What’s the difference between finding the lim to the left or to the right

When finding the limit of a function at a certain point, the difference between finding the **limit from the left** and the **limit from the right** is based on the direction from which we approach the point on the function's domain.

### 1. **Limit from the Left (Left-hand limit)**:
This is the limit of a function as the input approaches a specific value **from values less than** that point. It is written as:
\[
\lim_{x \to c^-} f(x)
\]
This means that as \(x\) approaches \(c\) from values **smaller** than \(c\), what value does \(f(x)\) approach?

### 2. **Limit from the Right (Right-hand limit)**:
This is the limit of a function as the input approaches the same value, but **from values greater than** that point. It is written as:
\[
\lim_{x \to c^+} f(x)
\]
This means that as \(x\) approaches \(c\) from values **larger** than \(c\), what value does \(f(x)\) approach?

### Key Point:
- **One-sided limits**: The left-hand and right-hand limits are referred to as "one-sided" limits. If both the left-hand and right-hand limits at a point exist and are equal, then the **two-sided limit** (just \(\lim_{x \to c} f(x)\)) exists and is equal to this common value.
  
- If \(\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)\), the **two-sided limit** does not exist at \(x = c\).

### Example:
Consider the function:
\[
f(x) = \begin{cases} 
      1 & \text{if } x < 0 \\
      2 & \text{if } x \geq 0 
   \end{cases}
\]
- Left-hand limit at \(x = 0\): 
\[
\lim_{x \to 0^-} f(x) = 1
\]
- Right-hand limit at \(x = 0\): 
\[
\lim_{x \to 0^+} f(x) = 2
\]

Since the left-hand and right-hand limits are not equal, \(\lim_{x \to 0} f(x)\) does not exist.

### Do you want more details or have any further questions?

### 5 related questions:
1. What is a **two-sided limit** and how does it relate to one-sided limits?
2. Can a function be continuous if the one-sided limits are different?
3. How do you interpret limits graphically on a piecewise function?
4. What happens if a limit approaches infinity from one side but a finite value from the other?
5. How do discontinuities affect one-sided limits?

### Tip:
Always check one-sided limits when dealing with piecewise or step functions to determine if a two-sided limit exists.

What’s the difference between finding the lim to the left or to the right?

Explore the difference between left-hand and right-hand limits in calculus. This explanation covers the basics of one-sided limits, helping students understand how limits work when approached from both directions. Ideal for high school students learning calculus.

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