Let f be defined as follows, where a does not = 0,
f(x) = {(x^2 - 2ax + 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.
Not too sure about this one and any help is appreciated!
Note that (x^2 - 2ax + a^2) is just (x-a)^2,
and hence f(x) when x =/= a is (x-a)^2/(x-a), which is (x-a)
This limit does exist as x approaches a.
f(a), also exists, because f(a) is f(x) when x = a, which is 5
However, f(x) is not continuous at x = a. This is because, f(a) = 5, as we have already seen. However, the limit of f(x) approaching a is given by (x-a), which tends to 0 at x approaches a.
So, the limit tending to a is not equal to f(a), and the function is not continuous at this point.
Hence, I and II are true, but III is not.
To determine which of the given statements are true about function f, 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 a for x in the function and simplify:
lim f(x) as x approaches a = lim [(x^2 - 2ax + a^2)/(x - a)] as x approaches a
Substituting x = a:
lim f(x) as x approaches a = [(a^2 - 2a^2 + a^2) / (a - a)] = 0/0
As we get an indeterminate form of 0/0, we can apply L'Hopital's Rule. Taking the derivative of the numerator and denominator and simplifying, we get:
lim f(x) as x approaches a = [(2a - 2a) / 1] = 0/1 = 0
Therefore, the limit exists and is equal to 0.
II. f(a) exists
To determine if f(a) exists, we substitute a for x in the given function:
f(a) = [(a^2 - 2a*a + a^2) / (a - a)] = [a^2 - 2a^2 + a^2] / 0
Since the denominator is 0, f(a) does not exist.
III. f(x) is continuous at x = a
For f(x) to be continuous at x = a, the limit as x approaches a should exist, and the function value at a should exist and be equal to the limit.
From our earlier calculation, we know that the limit as x approaches a is 0. However, since f(a) does not exist, f(x) is not continuous at x = a.
In conclusion, the correct answer is option E. I and II only.
To determine the answer to this question, we need to consider the properties of limits, the definition of a function being defined at a point, and the definition of continuity.
Let's go through each statement one by one.
I. lim f(x) exists as x approaches a.
To check the existence of the limit as x approaches a, we need to evaluate the limit of f(x) as x approaches a from both sides.
Let's consider the left-hand limit first:
lim f(x) as x approaches a- = lim [(x^2 - 2ax + a^2) / (x-a)] as x approaches a-
To evaluate this limit, we can simplify the expression by canceling the common factor (x-a) in both the numerator and denominator:
lim f(x) as x approaches a- = lim [(x-a)(x-a) / (x-a)] as x approaches a-
Now, we have:
lim f(x) as x approaches a- = lim (x-a) as x approaches a-
Similarly, let's consider the right-hand limit:
lim f(x) as x approaches a+ = lim [(x^2 - 2ax + a^2) / (x-a)] as x approaches a+
Again, we can simplify the expression:
lim f(x) as x approaches a+ = lim (x-a) as x approaches a+
Since both the left-hand limit and the right-hand limit are equal, we can conclude that the limit of f(x) as x approaches a exists.
II. f(a) exists.
To check if f(a) exists, we simply need to evaluate the function at x = a.
So, let's substitute x=a into the given function:
f(a) = [(a^2 - 2a(a) + a^2) / (a-a)] = [(a^2 - 2a^2 + a^2) / 0]
Here, we encounter a problem. The denominator is 0, which means the function is undefined at x=a. Therefore, f(a) does not exist.
III. f(x) is continuous at x = a.
A function is considered continuous at a point if three conditions hold:
1. The function is defined at that point (f(a) exists).
2. The limit of the function as x approaches that point exists (lim f(x) as x approaches a exists).
3. The value of the limit matches the value of the function at that point (f(a) = lim f(x) as x approaches a).
Since f(a) does not exist, the function fails to meet the condition of being defined at x=a. Therefore, f(x) is not continuous at x = a.
Based on the above analysis, the answer to the question is A. None.