The provided image shows the following mathematical expression:

$\lim_{{x \to 1}} \frac{2x^4 - 2x^3 - x^2 + 1}{x^4 - x^2 - 2x + 2}$

Let's solve this limit step by step.

Step 1: Direct Substitution

First, we'll substitute $x = 1$ directly into the expression to see if we get a determinate value.

$\text{Numerator at } x = 1: 2(1)^4 - 2(1)^3 - (1)^2 + 1 = 2 - 2 - 1 + 1 = 0$ $\text{Denominator at } x = 1: (1)^4 - (1)^2 - 2(1) + 2 = 1 - 1 - 2 + 2 = 0$

Since both the numerator and the denominator are $0$ when $x = 1$, we have a $\frac{0}{0}$ indeterminate form. Therefore, we can apply L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

L'Hôpital's Rule states that for limits of the form $\frac{0}{0}$, we can differentiate the numerator and the denominator separately and then take the limit.

Differentiating the numerator:

$\frac{d}{dx} \left( 2x^4 - 2x^3 - x^2 + 1 \right) = 8x^3 - 6x^2 - 2x$

Differentiating the denominator:

$\frac{d}{dx} \left( x^4 - x^2 - 2x + 2 \right) = 4x^3 - 2x - 2$

Step 3: Take the limit again

Now, we take the limit as $x \to 1$ of the new expression:

$\lim_{{x \to 1}} \frac{8x^3 - 6x^2 - 2x}{4x^3 - 2x - 2}$

Substitute $x = 1$:

$\text{Numerator: } 8(1)^3 - 6(1)^2 - 2(1) = 8 - 6 - 2 = 0$ $\text{Denominator: } 4(1)^3 - 2(1) - 2 = 4 - 2 - 2 = 0$

We still have a $\frac{0}{0}$ form, so we apply L'Hôpital's Rule again.

Step 4: Apply L'Hôpital's Rule again

Differentiate the numerator and denominator again.

Second derivative of the numerator:

$\frac{d}{dx} (8x^3 - 6x^2 - 2x) = 24x^2 - 12x - 2$

Second derivative of the denominator:

$\frac{d}{dx} (4x^3 - 2x - 2) = 12x^2 - 2$

Step 5: Take the limit one more time

Now, take the limit as $x \to 1$:

$\lim_{{x \to 1}} \frac{24x^2 - 12x - 2}{12x^2 - 2}$

Substitute $x = 1$:

$\text{Numerator: } 24(1)^2 - 12(1) - 2 = 24 - 12 - 2 = 10$ $\text{Denominator: } 12(1)^2 - 2 = 12 - 2 = 10$

Thus, the limit is:

$\boxed{1}$

Final Answer:

The value of the limit is $1$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How does L'Hôpital's Rule work and when can it be applied?
2. Can L'Hôpital's Rule be applied multiple times?
3. How do you differentiate polynomials step-by-step?
4. What other techniques are available for solving indeterminate forms?
5. How can we evaluate limits without using L'Hôpital's Rule?

Tip: Always check if direct substitution works first when solving limits before applying advanced methods like L'Hôpital's Rule.