Question

If f is the function defined by f(x) = (x - 9)/(x√(-3)), then lim(x → 9) f(x) is equivalent to which of the following?

a) [infinity]
b) -[infinity]
c) 0
d) Does not exist

Answer 1

The limit of the function f(x) as x approaches 9 does not exist. Therefore, the correct answer is: d) Does not exist

Step-by-step explanation:

To understand the limit of the function f(x) = (x - 9)/(x√(-3)) as x approaches 9, let's first examine the components of the function. Numerator: (x - 9).

This part is straightforward – as x approaches 9, the numerator approaches 0 because (9 - 9) = 0. Denominator: x√(-3). The square root of a negative number is not defined in the set of real numbers, as it results in an imaginary number.

The square root of -3 is an imaginary number, commonly expressed as √(-3) = i√3, where i is the imaginary unit. Therefore, for all values of x in the real numbers, the expression x√(-3) involves the imaginary unit i. Since we are concerned with the real number system, the function f(x) involves division by an imaginary number for any real value of x, which means the function is not defined for real numbers.

Now, let's analyze the limit: As x approaches 9 from either the left or the right, the numerator (x - 9) approaches 0.

However, since the denominator involves the imaginary unit (regardless of the value of x), the function does not have a real value at x=9 or in the neighborhood of x=9 in the real number system.

In conclusion, because the function involves division by an imaginary number, the limit of the function as x approaches 9 does not exist in the real number system. Therefore, the correct answer is: d) Does not exist