Determine the following limits.​
lim x→5 x − 7 / x(x − 5)^2

Verified step by step guidance

1. Identify the type of limit problem: This is a limit as \( x \) approaches 5, and the expression is \( \frac{x - 7}{x(x - 5)^2} \).
2. Check for direct substitution: Substitute \( x = 5 \) into the expression to see if it results in an indeterminate form. \( \frac{5 - 7}{5(5 - 5)^2} = \frac{-2}{0} \), which is undefined.
3. Recognize the indeterminate form: Since the denominator becomes zero, this is a case of an indeterminate form, suggesting the need for further analysis.
4. Consider simplifying or factoring: Since direct substitution leads to an undefined form, consider simplifying the expression or using algebraic manipulation to resolve the indeterminate form.
5. Apply L'Hôpital's Rule if applicable: If the limit results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule can be used by differentiating the numerator and the denominator separately and then taking the limit again.

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. In this case, we are interested in the limit as x approaches 5, which requires evaluating the function's behavior close to that point.

Indeterminate Forms

Indeterminate forms occur in limit problems when direct substitution leads to expressions like 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation, L'Hôpital's Rule, or other techniques to resolve the limit. In the given limit, substituting x = 5 results in an indeterminate form, necessitating additional steps to find the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, specifically 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule can simplify the process of finding limits, especially when direct evaluation is complex.