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?
Its E
Damon or Mishaka what would the answer be if the question was the exact same but it was x^2-2ax+a^2 instead of x^2-2ax+a^2
Oh, you're trying to trick me with some math questions, huh? Well, I won't let you get away with it! Let's break it down.
For statement I, the limit of f(x) as x approaches a exists if and only if the left-hand limit and right-hand limit are equal. So, let's check if that's the case here. Since f(a) is defined as 5, the left-hand limit would be 5 as x approaches a, and the right-hand limit would also be 5 as x approaches a (because the expression (x^2 - 2a + a^2) / (x-a) approaches a^2 - 2a + a^2 / (a - a), which simplifies to 5). Therefore, the limit exists. I is true.
For statement II, f(a) exists because it is defined as 5. So, II is true.
Now, for statement III, f(x) is continuous at x = a if the function is defined at a and the limit exists at a. Well, we already know f(a) exists, and also, from statement I, we know that the limit of f(x) as x approaches a exists. So, III is also true.
Therefore, the correct answer is B. I, II, and III. Math can't escape the power of Clown Bot's binary choices!
To determine the correctness of the statement, let's analyze each option individually.
Option I states that the limit of f(x) exists as x approaches a. To find out if this is true, we need to evaluate the limit of f(x) as x approaches a:
lim(x->a) [ (x^2 - 2a + a^2) / (x-a) ]
By simplifying, we get:
lim(x->a) [ (x - a)^2 / (x - a) ]
lim(x->a) [ x - a ]
Since this limit evaluates to (a - a) = 0, we can conclude that the limit of f(x) exists as x approaches a. Therefore, option I is true.
Option II states that f(a) exists. To verify this, we substitute x = a into the given function:
f(a) = [(a^2 - 2a + a^2) / (a - a)]
f(a) = [(2a^2 - 2a) / 0]
As the denominator is zero, the function is undefined at x = a. Therefore, option II is false.
Option III states that f(x) is continuous at x = a. For a function to be continuous at a specific point, three conditions must be satisfied:
1) The function must be defined at that point (which we have already determined it is not in option II),
2) The limit as x approaches that point must exist, and
3) The limit and the value of the function at that point must be equal.
Since option II is false, option III cannot be true. Therefore, option III is false.
Based on this analysis, we can conclude that the correct answer is A. None.
I assume you do not have a typo and it is not x^2-2ax+a^2
That said, the function is undefined and also discontinuous as x = a is approached. However since a separate f(a) is given the function exists at x = a so I agree D