We are given the limits:
$\lim_{x \to 1} f(x) = 1$ $\lim_{x \to 1} g(x) = -2$

We need to evaluate the following limit:

$\lim_{x \to 1} \left( \frac{2f(x) - f(x)}{3g(x) + 1} \right).$

Step 1: Substitute the Given Limits

Since we know the limits of $f(x)$ and $g(x)$, we substitute:

$\lim_{x \to 1} \left( \frac{2(1) - 1}{3(-2) + 1} \right).$

Step 2: Simplify the Expression

Calculate the numerator:

$2(1) - 1 = 2 - 1 = 1.$

Calculate the denominator:

$3(-2) + 1 = -6 + 1 = -5.$

Thus, the expression simplifies to:

$\frac{1}{-5} = -\frac{1}{5}.$

Final Answer:

$\lim_{x \to 1} \left( \frac{2f(x) - f(x)}{3g(x) + 1} \right) = -\frac{1}{5}.$

Would you like a more detailed explanation or have any questions?

Related Questions:

1. How do we evaluate limits involving fractions?
2. What are the conditions for a function’s limit to exist?
3. How can we apply L'Hôpital’s Rule if a limit results in an indeterminate form?
4. How do we handle limits involving piecewise functions?
5. What are one-sided limits, and how do they affect overall limits?

Tip: Always check if direct substitution works before attempting algebraic simplifications!