https://file.aiquickdraw.com/mathbot/file/1725499373845288ljz9m.jpg

Let's solve the given limit problem step by step. The problem is:

\[
\lim_{{x \to 1}} \frac{\frac{1}{{8 - 3x}} - \frac{1}{{11}}}{{\frac{1}{{6 - 3x}} - \frac{1}{{9}}}}
\]

### Step 1: Simplify the numerator and denominator

First, let's work on the numerator:
\[
\text{Numerator} = \frac{1}{{8 - 3x}} - \frac{1}{{11}}
\]
This can be rewritten using a common denominator:
\[
\text{Numerator} = \frac{11 - (8 - 3x)}{{11(8 - 3x)}} = \frac{3x - 8 + 11}{{11(8 - 3x)}} = \frac{3x + 3}{{11(8 - 3x)}}
\]
\[
\text{Numerator} = \frac{3(x + 1)}{{11(8 - 3x)}}
\]

Now, let's simplify the denominator:
\[
\text{Denominator} = \frac{1}{{6 - 3x}} - \frac{1}{{9}}
\]
This can also be rewritten using a common denominator:
\[
\text{Denominator} = \frac{9 - (6 - 3x)}{{9(6 - 3x)}} = \frac{3x - 6 + 9}{{9(6 - 3x)}} = \frac{3x + 3}{{9(6 - 3x)}}
\]
\[
\text{Denominator} = \frac{3(x + 1)}{{9(6 - 3x)}}
\]

### Step 2: Combine the results
Now, the entire expression becomes:
\[
\lim_{{x \to 1}} \frac{\frac{3(x + 1)}{{11(8 - 3x)}}}{{\frac{3(x + 1)}{{9(6 - 3x)}}}}
\]

We can simplify by canceling out the common factor \(3(x + 1)\) in both the numerator and denominator:
\[
\lim_{{x \to 1}} \frac{9(6 - 3x)}{11(8 - 3x)}
\]

### Step 3: Substitute \(x = 1\)
Substitute \(x = 1\) into the simplified expression:
\[
\frac{9(6 - 3(1))}{11(8 - 3(1))} = \frac{9(6 - 3)}{11(8 - 3)} = \frac{9 \times 3}{11 \times 5} = \frac{27}{55}
\]

Thus, the exact answer is:
\[
\boxed{\frac{27}{55}}
\]

Would you like further details on any of these steps, or do you have any questions?

Here are 5 related questions for further practice:
1. How would you solve \(\lim_{{x \to 2}} \frac{\frac{1}{{5 - 2x}} - \frac{1}{{7}}}{{\frac{1}{{3 - x}} - \frac{1}{{4}}}}\)?
2. What is the significance of common denominators in simplifying limit expressions?
3. How do you apply L'Hôpital's Rule to solve indeterminate forms in limits?
4. Can you find the limit \(\lim_{{x \to 0}} \frac{\sin(2x)}{x}\)?
5. What techniques can be used to evaluate limits that result in \(\frac{0}{0}\) indeterminate forms?

**Tip:** Always try to simplify complex fractions in limits by finding a common denominator or factoring common terms before substituting the limit value. This often avoids indeterminate forms and simplifies the evaluation process.

Evaluate lim as x approaches 1 of (1 / (8 - 3x) - 1 / 11) / (1 / (6 - 3x) - 1 / 9).

Learn how to evaluate the limit lim as x approaches 1 of a rational function using algebraic simplification. This step-by-step solution involves common denominator techniques and simplifying complex fractions. Suitable for students in Grades 10-12 or Calculus I.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation