We are tasked with finding the limit:

$\lim_{x \to c} (2x^3 - 3x^2 + x - 1)$

We will apply the properties of limits and the fact that the limit of a polynomial is simply the polynomial evaluated at the point $x = c$.

Step-by-Step Process:

1. Split the limit using the sum rule for limits:

The limit of a sum is the sum of the limits of the individual terms. $\lim_{x \to c} (2x^3 - 3x^2 + x - 1) = \lim_{x \to c} 2x^3 - \lim_{x \to c} 3x^2 + \lim_{x \to c} x - \lim_{x \to c} 1$

2. Apply the constant multiple rule:

The limit of a constant multiplied by a function is the constant multiplied by the limit of the function. For example: $\lim_{x \to c} 2x^3 = 2 \cdot \lim_{x \to c} x^3, \quad \lim_{x \to c} 3x^2 = 3 \cdot \lim_{x \to c} x^2$

3. Use the power rule for limits:

The limit of a power of a function is the power of the limit of the function. Specifically, the following hold: $\lim_{x \to c} x^3 = c^3, \quad \lim_{x \to c} x^2 = c^2, \quad \lim_{x \to c} x = c$

4. Use the constant rule:

The limit of a constant is just the constant itself: $\lim_{x \to c} 1 = 1$

5. Substituting all values:

Now, substituting these values back into the expression: $\lim_{x \to c} (2x^3 - 3x^2 + x - 1) = 2c^3 - 3c^2 + c - 1$

Thus, the limit is: $\boxed{2c^3 - 3c^2 + c - 1}$

Summary:

The limit of the polynomial $2x^3 - 3x^2 + x - 1$ as $x \to c$ is $2c^3 - 3c^2 + c - 1$.

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Would you like more details or have any questions?

Here are 5 follow-up questions:

1. How does the constant multiple rule help in solving polynomial limits?
2. Can the same approach be applied to rational functions?
3. What would the limit be if the polynomial was of higher degree, say $2x^5$?
4. How does the power rule simplify the limit calculation of powers of $x$?
5. What is the significance of limits in calculus and continuous functions?

Tip: For polynomials, the limit at a point is just the evaluation of the polynomial at that point!