Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x² − 3x + 2) / (x³ − 2x²) as


d. x→2

Verified step by step guidance

1. First, identify the type of limit problem. This is a rational function limit as x approaches a specific value, x = 2.
2. Check if direct substitution of x = 2 into the function results in an indeterminate form. Substitute x = 2 into the numerator and denominator: numerator becomes 2² - 3(2) + 2 and denominator becomes 2³ - 2(2²).
3. Calculate the values from the substitution: numerator becomes 4 - 6 + 2 = 0 and denominator becomes 8 - 8 = 0. This results in the indeterminate form 0/0.
4. To resolve the indeterminate form, factor both the numerator and the denominator. The numerator x² - 3x + 2 factors into (x - 1)(x - 2) and the denominator x³ - 2x² factors into x²(x - 2).
5. Cancel the common factor (x - 2) from the numerator and denominator, resulting in the simplified expression (x - 1)/x². Now, substitute x = 2 into the simplified expression to find the limit.

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given limit problem, both the numerator and denominator are polynomials, which can lead to specific behaviors as x approaches certain values. Understanding how to simplify rational functions is essential for evaluating limits, especially when direct substitution results in indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require further analysis, such as factoring, simplifying, or applying L'Hôpital's Rule to resolve the limit correctly.