lim
x→ 1
2
x−2 − x−1 − 2
2x − 1

To find the limit of the given function as \( x \) approaches 1, we have:

\[
\lim_{{x \to 1}} \left( \frac{2}{{x-2}} - \frac{x-1}{2x-1} \right)
\]

First, we should check the behavior of each term separately as \( x \) approaches 1.

### For the first term: \(\frac{2}{x-2}\)

As \( x \) approaches 1, the denominator \( x-2 \) approaches -1. Thus:
\[
\frac{2}{x-2} \rightarrow \frac{2}{1-2} = \frac{2}{-1} = -2
\]

### For the second term: \(\frac{x-1}{2x-1}\)

As \( x \) approaches 1, the denominator \( 2x-1 \) approaches 1. Thus:
\[
\frac{x-1}{2x-1} \rightarrow \frac{1-1}{2 \cdot 1-1} = \frac{0}{1} = 0
\]

### Combining both terms:

Now we combine these results to find the limit of the entire expression:
\[
\lim_{{x \to 1}} \left( \frac{2}{{x-2}} - \frac{x-1}{2x-1} \right) = -2 - 0 = -2
\]

Thus, the limit is:
\[
\boxed{-2}
\]

Would you like any additional details or have any questions about this solution?

Here are some related questions to consider:

1. How does the behavior of a function near a point help determine its limit?
2. What is the significance of the numerator approaching zero in the second term?
3. How can you determine if a function is continuous at a point?
4. What are some common techniques for finding limits involving fractions?
5. Why does the first term approach -2 as \( x \) approaches 1?
6. What happens if a limit does not exist at a particular point?
7. How can L'Hôpital's Rule be applied to find limits?
8. What is the difference between a limit approaching infinity and a limit approaching a finite value?

**Tip:** Always simplify the function as much as possible before evaluating the limit to make the calculations easier.

Learn how to calculate the limit of the rational function (2/(x-2)) - ((x-1)/(2x-1)) as x approaches 1. This problem involves understanding limit laws and applying them to rational functions. Suitable for college-level mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation