https://file.aiquickdraw.com/mathbot/file/1727451933903i6od5nn2.jpg

The problem is asking you to evaluate the following limit:

\[
\lim_{{x \to 1}} \frac{{x - 6}}{{x^2 - 3x - 18}}
\]

Additionally, you're asked to create a table of values for the function \( f(x) = \frac{{x - 6}}{{x^2 - 3x - 18}} \) for values of \(x\) close to 1. The function is discontinuous at \(x = 1\), so estimating the limit numerically is a good approach.

### Let's break this down:

1. **Factor the denominator**: 
   The denominator is \( x^2 - 3x - 18 \), which we can factor:
   \[
   x^2 - 3x - 18 = (x - 6)(x + 3)
   \]

   Now, the function becomes:
   \[
   f(x) = \frac{{x - 6}}{{(x - 6)(x + 3)}}
   \]
   For \( x \neq 6 \), this simplifies to:
   \[
   f(x) = \frac{1}{{x + 3}}
   \]
   But note that this simplification holds only when \( x \neq 6 \), so you can use this function to fill in the table for \( x \) values near 1.

2. **Create the table**:
   I'll now calculate the function \( f(x) = \frac{1}{x+3} \) for the given \(x\) values (0.9, 0.99, 0.999, 1.001, 1.01, and 1.1).

Let me calculate these values and fill in the table for you.Here are the values for the function \( f(x) = \frac{1}{x + 3} \), rounded to four decimal places:

| \(

Consider the following limit: lim x→1 (x−6) / (x^2 − 3x − 18). Create a table of values for f(x) for x close to 1 and estimate the limit.

Learn how to estimate the limit of the function (x−6)/(x^2−3x−18) as x approaches 1. Includes factoring the quadratic expression and creating a table of values. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation