Question

If f is the function defined by f(x)=(1/x−1)/(x−1), then limx→1f(x) is equivalent to which of the following?

A) 0
B) 1
C) [infinity]
D) Does not exist

Answer 1

The limit of the function f(x) as x approaches 1 is -1. This can be determined by simplifying the function and applying the limit. The options provided in the question do not include the correct answer, indicating a potential error in the question.

Step-by-step explanation:

To evaluate the limit of f(x) as x approaches 1, where f(x) is given by the function f(x) = (1/x - 1) / (x - 1), we need to understand the behavior of the function as it approaches the given point. Direct substitution of x = 1 gives us an indeterminate form 0/0, which means we need to apply algebraic manipulation or L'Hôpital's Rule to resolve this form and compute the actual limit. By factoring and simplification, we can rewrite the function f(x) as:

f(x) = ((1 - x) / x) / (x - 1) = -1 / x

Now, if we take the limit as x approaches 1, we get:

lim(x→1) f(x) = lim(x→1) (-1 / x) = -1 / 1 = -1

Therefore, the limit of f(x) as x approaches 1 is -1, which is not one of the options listed (A, B, C, D) in the original question. It appears there may be a mistake in the question or the provided options, as none of them match the correct answer.