Let f be defined as follows, where a ≠ 0, f(x) = { (x^2-2ax+a^2)/x-a if x ≠ a, 5 if x = a
Which of the following are true about f?
I. lim(fx) exists
x --> a
II. f(a) exists
III. f(x) is continuous at x = a
Thank you in advance!
f(x) = (x^2-2ax+a^2)/(x-a) , notice the important bracket I inserted for (x-a)
lim (x^2-2ax+a^2)/(x-a) , as x ---> a
= lim (x-a)^2/(x-a) , as x ---> a
= lim x-a, as x ---> 0
= a-a = 0
draw your conclusions
just consider.
For all x ≠ a, f(x) = x-a
To determine the truth about the statements I, II, and III, let's analyze each of them separately.
I. lim(fx) exists as x approaches a:
To find the limit as x approaches a, we substitute x = a into the function:
lim(x --> a) [(x^2 - 2ax + a^2)/(x - a)]
lim(x --> a) [(a^2 - 2a*a + a^2)/(a - a)]
lim(x --> a) [(a^2 - 2a^2 + a^2)/(0)]
lim(x --> a) [-a^2/0]
The limit does not exist since it yields an indeterminate form (0/0). Therefore, statement I is false.
II. f(a) exists:
To find f(a), we substitute x = a into the function:
f(a) = [(a^2 - 2a*a + a^2)/(a - a)] = 0/0
The function is undefined at x = a, which means f(a) does not exist. Therefore, statement II is false.
III. f(x) is continuous at x = a:
A function is continuous at a point if the limit as x approaches that point exists and is equal to the function value at that point.
Since the limit does not exist (as mentioned in statement I) and f(a) does not exist (as mentioned in statement II), f(x) is not continuous at x = a.
Therefore, statement III is false.
In summary, none of the statements I, II, and III are true about f.
To determine which of the statements about f are true, we need to analyze the definitions and properties of limits, continuity, and function evaluation.
I. To check if the limit of f(x) exists as x approaches a, we evaluate the limit from both sides of a. In this case, we have:
lim(x → a-) f(x) = lim(x → a-) ((x^2 - 2ax + a^2)/(x - a))
= lim(x → a-) ((x - a)(x - a)/(x - a))
= lim(x → a-) (x - a)
= 0
lim(x → a+) f(x) = lim(x → a+) ((x^2 - 2ax + a^2)/(x - a))
= lim(x → a+) ((x - a)(x - a)/(x - a))
= lim(x → a+) (x - a)
= 0
Since the limit from both sides of a is equal to 0, the limit of f(x) as x approaches a exists and is equal to 0. Therefore, statement I is true.
II. To evaluate f(a), we substitute the value of a into the function:
f(a) = ((a^2 - 2a*a + a^2)/(a - a)) = (0/0)
Here, we have an indeterminate form of division by zero. So, f(a) does not exist. Therefore, statement II is false.
III. To check if f(x) is continuous at x = a, we need to verify that the limit of f(x) as x approaches a is equal to the function value at a. In this case, we have:
lim(x → a) f(x) = lim(x → a) ((x^2 - 2ax + a^2)/(x - a))
= lim(x → a) ((x - a)(x - a)/(x - a))
= lim(x → a) (x - a)
= 0
Since lim(x → a) f(x) = 0, and f(a) does not exist, f(x) is not continuous at x = a. Therefore, statement III is false.
In summary, statement I is true, statement II is false, and statement III is false.