Let f be defined as follows, where a does not = 0,
f(x) = {(x^2 - 2a + a^2) / (x-a), if x does not = a
5, if x = a
Which of the following are true about f?
I. lim f(x) exists as x approaches a
II. f(a) exists
III. f(x) is continuous at x = a.
A. None
B. I, II, and III
C. I only
D. II only
E. I and II only.
From my own knowledge, I would say that it is D. II only. Since we do not know what is equal to, we cannot determine what the limits or continuity would be at a. Is this correct?
To determine which of the given statements about f are true, let's analyze each statement one by one.
I. lim f(x) exists as x approaches a
To find the limit as x approaches a, we can substitute x with a in the definition of f:
lim f(x) as x approaches a = lim ((x^2 - 2a + a^2) / (x-a)) as x approaches a
Since the numerator and denominator both become zero when x = a, this limit is an indeterminate form of 0/0. Therefore, we can apply L'HΓ΄pital's rule:
lim f(x) as x approaches a = lim (2x - 2a) / 1 = 2a - 2a = 0.
So, the limit of f(x) as x approaches a exists and is equal to zero. Thus, statement I is true.
II. f(a) exists
To find f(a), we substitute x = a into the definition of f:
f(a) = (a^2 - 2a + a^2) / (a-a) = (2a^2 - 2a) / 0.
Since division by zero is undefined, f(a) does not exist. Thus, statement II is false.
III. f(x) is continuous at x = a
A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the value of the function at that point.
As we found in statement I, the limit of f(x) as x approaches a is 0. However, as we determined in statement II, f(a) does not exist. Therefore, the limit of f(x) as x approaches a is not equal to the value of f(a), which means f(x) is not continuous at x = a. Thus, statement III is false.
So, the correct answer is A. None of the statements I, II, and III are true about f.