Calculate the limits
lim x->-1

2x-1/4x^2+5x-2

To calculate the limit of the function \( \frac{2x - 1}{4x^2 + 5x - 2} \) as \( x \) approaches \(-1\), we follow these steps:

1. **Substitute \( x = -1 \) into the function:**
   \[
   \frac{2(-1) - 1}{4(-1)^2 + 5(-1) - 2}
   \]
   Simplifying the numerator:
   \[
   2(-1) - 1 = -2 - 1 = -3
   \]
   Simplifying the denominator:
   \[
   4(-1)^2 + 5(-1) - 2 = 4(1) - 5 - 2 = 4 - 5 - 2 = -3
   \]

2. **Evaluate the function:**
   \[
   \frac{-3}{-3} = 1
   \]

Thus, the limit is:
\[
\lim_{x \to -1} \frac{2x - 1}{4x^2 + 5x - 2} = 1
\]

Would you like more details or have any questions about the solution?

### Relative Questions:
1. How do you find the limit of a function as \( x \) approaches a particular value?
2. What happens if substituting the value results in an indeterminate form like \( \frac{0}{0} \)?
3. Can you explain the concept of continuity in the context of limits?
4. How do you apply L'Hôpital's Rule to solve limit problems?
5. What are the different techniques for evaluating limits of rational functions?
6. How does factoring help in finding the limits of rational functions?
7. What are one-sided limits and how do they differ from two-sided limits?
8. Can limits be used to determine the behavior of a function at infinity?

### Tip:
When evaluating limits, always start by direct substitution to check if the function is continuous at that point. If not, explore algebraic simplification, factoring, or L'Hôpital's Rule to find the limit.

Learn how to calculate the limit of the function (2x - 1) / (4x^2 + 5x - 2) as x approaches -1. Follow a detailed step-by-step solution, including substitution and simplification techniques. Suitable for high school students in Grades 11-12 studying limits and rational functions.

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