60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 


lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

Verified step by step guidance

1. First, substitute x = 1 into the function to check if the limit results in an indeterminate form. Calculate the numerator: 1^4 - 1^3 - 3(1)^2 + 5(1) - 2 and the denominator: 1^3 + 1^2 - 5(1) + 3.
2. After substitution, if both the numerator and denominator evaluate to 0, the limit is in the indeterminate form 0/0, which means l'Hôpital's Rule can be applied.
3. Apply l'Hôpital's Rule by differentiating the numerator and the denominator separately. The derivative of the numerator (x^4 - x^3 - 3x^2 + 5x - 2) is 4x^3 - 3x^2 - 6x + 5.
4. The derivative of the denominator (x^3 + x^2 - 5x + 3) is 3x^2 + 2x - 5.
5. Evaluate the limit of the new function formed by the derivatives as x approaches 1: lim_x→1 (4x^3 - 3x^2 - 6x + 5) / (3x^2 + 2x - 5). Substitute x = 1 into this expression to find the limit.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 1 involves determining the behavior of the given rational function near that point.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex rational functions.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given limit problem, both the numerator and denominator are polynomials. Understanding their behavior, such as factoring or finding roots, is crucial for evaluating limits effectively.