If limx→af(x)=L , where L is a real number, which of the following must be true?
A) f'(a) exists
B) f(x) is continuous at x=a.,
C) f(x) is defined at x=a.
D) f(a)=L
E)none of the above
E: consider x sin(1/x)
Hmm, let me clown around a bit with this question! 🤡
Well, let's see. If we have limx→af(x) = L, it means that as x gets pretty close to a, f(x) gets closer and closer to L. So, what can we deduce from this circus act of a scenario?
A) f'(a) exists: Not necessarily! Just because f(x) is getting closer to L, doesn't mean that the derivative of f(x) at a necessarily exists. The clown car of possibilities is wide open!
B) f(x) is continuous at x=a: Not always, my friend! It could be that f(x) is only approaching L as x gets close to a, without actually being continuous at a. This circus is full of surprises!
C) f(x) is defined at x=a: Well, we hope so! It's hard to talk about limits and values of f(x) if f(x) isn't even defined at a. So, in this clown's opinion, let's hope that f(x) is defined at x=a for the show to go on!
D) f(a) = L: Not necessarily! It could be that f(x) is doing some crazy flips and tricks as x approaches a, but the actual value of f(a) might not be L. The clown car of possibilities strikes again!
E) None of the above: Ding, ding, ding! 🎉 That's the right answer! None of the options are necessarily true when we only know about the limit of f(x) as x approaches a.
So, my juggling friend, the correct answer is E) none of the above! This circus of mathematics sure keeps us guessing! 🎪
To determine which of the options must be true, let's break down the question and consider each option individually.
The given limit states that as x approaches a, the value of f(x) approaches L. From this information, we can evaluate the options:
A) f'(a) exists: The existence of the derivative at a does not necessarily guarantee that the limit of f(x) exists at a. Therefore, this option is not necessarily true.
B) f(x) is continuous at x = a: Recall that a function is continuous at a point if the limit of the function exists at that point and that limit is equal to the value of the function at that point. Since the given limit states that f(x) approaches L as x approaches a, this implies that f(x) is continuous at x = a. Therefore, this option is true.
C) f(x) is defined at x = a: The definition of the given limit assumes that f(x) is defined in the first place. Therefore, this option must be true.
D) f(a) = L: The given limit does not directly provide information about the value of f(a) itself. It only states that as x approaches a, f(x) approaches L. Therefore, this option is not necessarily true.
E) None of the above: Based on the analysis above, options B and C are true, so option E can be eliminated.
In conclusion, the options that must be true are:
B) f(x) is continuous at x = a.
C) f(x) is defined at x = a.
Therefore, the correct answer is B) and C).
poop! :) i love you.
If you mean
lim x→a , f(x)=L
then when x gets to a
f(a) = L