Determine the following limits.


c. lim x→4 x − 5 / (x − 4)^2

Verified step by step guidance

1. Step 1: Identify the type of limit problem. This is a limit as x approaches a specific value, which may involve an indeterminate form. We need to evaluate \( \lim_{{x \to 4}} \frac{{x - 5}}{{(x - 4)^2}} \).
2. Step 2: Substitute x = 4 into the expression to check for indeterminate form. Substituting gives \( \frac{{4 - 5}}{{(4 - 4)^2}} = \frac{{-1}}{{0}} \), which is undefined, indicating a potential vertical asymptote or infinite limit.
3. Step 3: Analyze the behavior of the function as x approaches 4 from the left (x → 4⁻) and from the right (x → 4⁺). Consider the sign of the numerator and denominator in each case to determine the direction of the limit.
4. Step 4: For x approaching 4 from the left (x → 4⁻), the numerator (x - 5) is negative, and the denominator (x - 4)^2 is positive, leading to a negative result. For x approaching 4 from the right (x → 4⁺), the numerator is still negative, and the denominator remains positive, also leading to a negative result.
5. Step 5: Conclude that since both one-sided limits approach negative infinity, the overall limit \( \lim_{{x \to 4}} \frac{{x - 5}}{{(x - 4)^2}} \) is negative infinity.

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 4, which requires evaluating the function's behavior close to that point.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to an ambiguous result, such as 0/0 or ∞/∞. In the given limit, substituting x = 4 results in the form 0/0, indicating that further analysis is needed to determine the limit. Techniques such as factoring, rationalizing, or applying L'Hôpital's Rule can be used to resolve these forms.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms. It states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits in cases where direct evaluation is not possible.