Question

Let f be a function that is defined for all real numbers x. of the following which is the best interpretation of the statement lim f(x) =5 ?

Answer 1

The statement 'lim f(x) = 5' indicates that the function f(x) approaches the value 5 as x gets closer to a particular point. In the context of continuous probability functions, this concept relates to how probabilities are represented by the areas under probability density functions.

Step-by-step explanation:

Understanding the Limit of a Function

The statement lim f(x) = 5 implies that as the input values x approach a certain point from either side, the function f(x) approaches the value 5. In other words, as x gets closer and closer to this point (which may be a specific value or infinity), the output of the function f(x) becomes arbitrarily close to 5. This concept is fundamental in calculus and is used to describe the behavior of functions as inputs approach a given value.

When we consider the function f(x) being a continuous probability function or probability density function (PDF), the interpretation of limits also applies. However, in probability theory, the areas under the PDF represent probabilities. The maximum area under a PDF over its entire range is 1, which corresponds to the certain event with a probability of 100%.

If we have a continuous probability distribution between 0 and a specific value, asking for the probability that x equals a specific value within that range will always result in a probability of zero because the area at a single point is zero in a continuous distribution. Instead, we look at intervals, such as P(2.5 < x < 7.5), which is found by calculating the area under the PDF between the two values.

Answer 2

As x approaches 3, the value of f(x) approaches 5. Option C is correct.

Explanations:

The definition of limit of a function tells us that the limit of a function equals a number as the input value approaches or move close to a value.

For the given limit of a function

`\lim_(x\to3)f(x)=5`

Shows that the limit of the function is 5 as the input value of "x" is closer or approaches 3.

Hence the best interpretation of the statement is that as x approaches 3, the value of f(x) approaches 5