Determine the following limits.


a. lim x→4^+ x − 5 / (x − 4)^2

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 of discontinuity or infinity. In this case, we are interested in the limit as x approaches 4 from the right (denoted as x→4^+).

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The notation x→4^+ indicates that we are considering values of x that are greater than 4. This is crucial for analyzing functions that may behave differently when approaching a point from the left versus the right.

Indeterminate Forms

Indeterminate forms occur in limit problems when direct substitution leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this question, substituting x=4 into the expression results in an indeterminate form, necessitating further analysis, such as factoring or applying L'Hôpital's Rule to resolve the limit.