https://file.aiquickdraw.com/mathbot/file/17270396080251kz75i5q.jpg

The function given is:

\[
f(x) = \frac{x^2 - 1}{x - 1}
\]

### Step 1: Simplify the Function
Notice that \( x^2 - 1 \) is a difference of squares, which can be factored as:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

So the function becomes:

\[
f(x) = \frac{(x - 1)(x + 1)}{x - 1}
\]

For \( x \neq 1 \), we can cancel the common term \( (x - 1) \), which simplifies the function to:

\[
f(x) = x + 1 \quad \text{for} \quad x \neq 1
\]

### Step 2: Analyze the Limit

Now, we need to find the limit:

\[
\lim_{x \to 1} f(x)
\]

Since the simplified function is \( f(x) = x + 1 \) for \( x \neq 1 \), we can directly substitute \( x = 1 \) into the simplified function:

\[
\lim_{x \to 1} (x + 1) = 1 + 1 = 2
\]

Thus, the limit exists and is:

\[
\lim_{x \to 1} f(x) = 2
\]

### Step 3: Behavior at \( x = 1 \)

Even though the limit as \( x \to 1 \) exists, the function is undefined exactly at \( x = 1 \) because the original function has the term \( \frac{0}{0} \) at this point, which is indeterminate. Thus, there is a **hole** in the graph at \( x = 1 \).

### Step 4: Graph of the Function

- For \( x \neq 1 \), the function behaves like a straight line \( f(x) = x + 1 \).
- At \( x = 1 \), there is a hole in the graph.

The graph will be a straight line with a slope of 1 and a y-intercept at \( (0, 1) \), but with a hole at the point \( (1, 2) \).

### Summary

- The limit \( \lim_{x \to 1} f(x) = 2 \) exists.
- The function is undefined at \( x = 1 \), so there is a hole in the graph at \( (1, 2) \).

Would you like a detailed graph or further clarification on any part?

---

Here are 5 related questions:

1. What happens to the limit of a function if the denominator does not factor cleanly?
2. How do you handle limits when a function gives a \( \frac{0}{0} \) indeterminate form?
3. Can limits exist if a function is undefined at a point?
4. How does factoring help in simplifying rational functions for limits?
5. What is the difference between one-sided and two-sided limits?

**Tip:** Always check for factoring opportunities when simplifying functions to resolve indeterminate forms.

Graph the function f(x) = (x^2 - 1) / (x - 1) and find lim(x→1) f(x). Use your graph to help find the limit and discuss whether the limit exists.

This problem focuses on graphing the function f(x) = (x^2 - 1) / (x - 1) and finding the limit as x approaches 1. It covers factorization techniques and resolving indeterminate forms, specifically how to handle limits of functions that yield 0/0. A detailed step-by-step solution is provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation