If a function f(x) is undefined at x=a, then the limit f(x) must not exist.
True or False?
false.
f(x) = (x-a)/(x-a)
Looks like f(x) = 1, and it does -- everywhere but at x=a, where it is 0/0.
Well, isn't that a tricky question? I guess it depends on how you define "undefined." If the function is truly undefined at x = a, then it wouldn't make sense to talk about the limit at that point. So, in that case, I would say it's true! But hey, don't fret if you're unsure, math can be quite the circus sometimes!
False. If a function f(x) is undefined at x=a, it does not necessarily mean that the limit f(x) does not exist. The limit may still exist, but the function may have a removable or non-removable discontinuity at x=a. The existence of the limit depends on the behavior of the function as x approaches a from both sides.
True.
To understand why this statement is true, let's start by explaining the concept of a limit. The limit of a function f(x) as x approaches a (written as lim[x->a] f(x)) is a fundamental concept in calculus. It represents the value that f(x) approaches as x gets arbitrarily close to a.
Now, if a function f(x) is undefined at x=a, it means that the function does not have a value specifically at that point. This could happen, for example, if there is a discontinuity or a hole in the graph of the function at x=a.
If the function f(x) does not have a value at x=a, it becomes impossible to determine what value f(x) would approach as x approaches a. In other words, the limit does not exist. This is because the existence of a limit requires that the function has a value or the function's behavior can be determined as x gets closer to the given point.
Therefore, if a function f(x) is undefined at x=a, it implies that the limit f(x) does not exist. Hence, the statement is true.