lim*𝑥*→14−4*𝑥*−1*𝑥*−2+*𝑥*−1=limx→14−4x−1x−2+x−1=

To find the limit

\[
\lim_{x \to 14^-} \left( \frac{-4x - 1}{x - 2} + \frac{1}{x - 1} \right)
\]

we need to analyze the behavior of the function as \(x\) approaches 14 from the left. Let's simplify and solve step-by-step.

### Step-by-Step Solution

1. **Combine the fractions:**

   The expression involves two fractions. To combine them, we need a common denominator, which will be \((x - 2)(x - 1)\):

   \[
   \frac{-4x - 1}{x - 2} + \frac{1}{x - 1} = \frac{(-4x - 1)(x - 1) + (x - 2)}{(x - 2)(x - 1)}
   \]

2. **Expand the numerator:**

   Expand the terms in the numerator:

   \[
   = \frac{(-4x - 1)(x - 1) + (x - 2)}{(x - 2)(x - 1)}
   \]

   Now distribute and simplify:

   \[
   = \frac{(-4x^2 + 4x - x + 1) + (x - 2)}{(x - 2)(x - 1)}
   \]

   Combine like terms:

   \[
   = \frac{-4x^2 + 4x - x + 1 + x - 2}{(x - 2)(x - 1)}
   \]

   \[
   = \frac{-4x^2 + 4x - x + x + 1 - 2}{(x - 2)(x - 1)}
   \]

   \[
   = \frac{-4x^2 + 4x}{(x - 2)(x - 1)}
   \]

   \[
   = \frac{-4x^2 + 4x}{(x - 2)(x - 1)}
   \]

3. **Factor the numerator:**

   Factor out a common term in the numerator:

   \[
   = \frac{-4x(x - 1)}{(x - 2)(x - 1)}
   \]

   Cancel out the \((x - 1)\) term, since \(x \neq 1\):

   \[
   = \frac{-4x}{x - 2}
   \]

4. **Evaluate the limit:**

   Now, we evaluate the limit as \(x \to 14^-\):

   \[
   \lim_{x \to 14^-} \frac{-4x}{x - 2}
   \]

   Substituting \(x = 14\):

   \[
   = \frac{-4(14)}{14 - 2} = \frac{-56}{12} = -\frac{14}{3}
   \]

### Conclusion

The value of the limit is:

\[
\lim_{x \to 14^-} \left( \frac{-4x - 1}{x - 2} + \frac{1}{x - 1} \right) = -\frac{14}{3}
\]

Would you like any further explanation on a specific step?

### Related Questions

1. How can limits approaching infinity or a certain point be evaluated?
2. What is the importance of checking for discontinuities in rational functions?
3. How do you find one-sided limits involving fractions?
4. What are the rules for combining fractions with different denominators?
5. How do you use algebraic manipulation to simplify complex expressions?

**Tip:** Simplifying expressions by combining fractions with common denominators is often the key step in evaluating limits.

Learn how to evaluate the limit lim(x → 14^-) ((-4x - 1)/(x - 2) + 1/(x - 1)). This problem involves combining fractions with different denominators and evaluating a limit involving rational functions. Suitable for advanced high school students and calculus enthusiasts.

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