Determine the following limits.


$\underset{x\to 0}{\lim }\frac{x^3-5x^2}{x^2}$

Verified step by step guidance

1. Step 1: Identify the limit expression: \( \lim_{x \to 0} \frac{x^3 - 5x^2}{x^2} \).
2. Step 2: Simplify the expression by factoring out the common term in the numerator. The numerator \( x^3 - 5x^2 \) can be factored as \( x^2(x - 5) \).
3. Step 3: Cancel the common factor \( x^2 \) in the numerator and the denominator. This simplifies the expression to \( x - 5 \).
4. Step 4: Substitute \( x = 0 \) into the simplified expression \( x - 5 \) to evaluate the limit.
5. Step 5: Conclude the limit by substituting the value of \( x \) into the simplified expression.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. For example, the limit of a function as x approaches 0 can reveal the function's value or behavior at that point, even if the function itself is not explicitly defined there.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In the context of limits, factoring can simplify complex rational expressions, making it easier to evaluate limits by canceling out common terms. For instance, in the limit problem given, factoring the numerator can help eliminate the indeterminate form that arises when substituting the limit directly.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not provide clear information about the limit's value, such as 0/0 or ∞/∞. These forms require further analysis, often through algebraic manipulation, L'Hôpital's Rule, or other techniques, to resolve. Recognizing an indeterminate form is crucial for applying the appropriate methods to find the actual limit.