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The given expression is:

\[
\lim_{{x \to 1}} \frac{x^4 - 3x + 2}{x^5 - 4x + 3}
\]

To find this limit, let's evaluate the function at \( x = 1 \):

1. **Numerator at \( x = 1 \)**:
   \[
   x^4 - 3x + 2 \quad \text{at} \quad x = 1: \quad 1^4 - 3(1) + 2 = 1 - 3 + 2 = 0
   \]

2. **Denominator at \( x = 1 \)**:
   \[
   x^5 - 4x + 3 \quad \text{at} \quad x = 1: \quad 1^5 - 4(1) + 3 = 1 - 4 + 3 = 0
   \]

Since both the numerator and denominator are 0 when \( x = 1 \), this is an indeterminate form \(\frac{0}{0}\), so we can apply **L'Hopital's Rule**, which requires us to differentiate the numerator and the denominator and then re-evaluate the limit.

### Differentiating the numerator:
\[
\frac{d}{dx}\left( x^4 - 3x + 2 \right) = 4x^3 - 3
\]

### Differentiating the denominator:
\[
\frac{d}{dx}\left( x^5 - 4x + 3 \right) = 5x^4 - 4
\]

Now, re-evaluate the limit using the derivatives:

\[
\lim_{{x \to 1}} \frac{4x^3 - 3}{5x^4 - 4}
\]

Substitute \( x = 1 \) into the derivatives:

1. **Numerator at \( x = 1 \)**:
   \[
   4(1)^3 - 3 = 4 - 3 = 1
   \]

2. **Denominator at \( x = 1 \)**:
   \[
   5(1)^4 - 4 = 5 - 4 = 1
   \]

Thus, the limit becomes:
\[
\frac{1}{1} = 1
\]

Therefore, the limit is:

\[
\boxed{1}
\]

Would you like more details or have any questions?

Here are some related questions to explore further:
1. What is L'Hopital's Rule and when can it be applied?
2. Can this limit be evaluated using another method besides L'Hopital's Rule?
3. How can you determine if a function has a removable discontinuity at a given point?
4. What are other examples of indeterminate forms, and how can they be resolved?
5. How can polynomial division help in evaluating limits?

**Tip:** Always check for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) before applying L'Hopital's Rule.

lim (x → 1) (x^4 - 3x + 2) / (x^5 - 4x + 3)

Learn how to evaluate the limit lim (x → 1) (x^4 - 3x + 2) / (x^5 - 4x + 3) using L'Hopital's Rule. This problem involves indeterminate forms and requires differentiation of polynomial functions. Suitable for advanced high school or college calculus students.

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