Final answer:
Finding the limit when x approaches 4 from the right means examining the behavior of a function as x gets arbitrarily close to 4 from values greater than 4, not the behavior as x approaches positive infinity or the average rate of change. It involves looking at increasingly small intervals to the right of 4, to understand the function's behavior near that point. Option C is the correct answer.
Step-by-step explanation:
When we talk about finding the limit of a function as x approaches a particular value, we're typically dealing with a mathematical concept that describes the behavior of a function as the input value (x) gets infinitely close to a certain number. In this case, the student is asking specifically what it means to find the limit when x approaches 4 from the right, which is option C: Examining the behavior of the function as x gets arbitrarily close to 4 from values greater than 4.
This concept is particularly important in situations where the function may not be defined at x = 4 or where the function exhibits interesting behaviors around that point. For example, using the function y = 1/x, which was provided in the reference information, we can see that the function approaches infinity as x approaches 0. This is an example of a function having an asymptote. However, for the limit as x approaches 4 from the right, we would be looking at values of x such as 4.1, 4.01, 4.001 and so forth, slowly getting closer to 4 without actually ever reaching it.
It is different from the behavior of the function as x approaches positive infinity or the average rate of change. It is also different from evaluating y-values as x approaches 4 from the left, which would look at values just less than 4, such as 3.9, 3.99, 3.999, and so on.
An important note to make is that sometimes the limit from the right does not equal the limit from the left. If these two are equal, we say the function is continuous at that point. If they are not equal, the function has a discontinuity at that point. Understanding and finding limits is an essential skill in calculus, as it lays the foundation for defining derivatives and integrals, which describe rates of change and accumulation, respectively.