Determine whether the following statements are true and give a...

Question

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

If limx→a f(x) = L, then f(a)=L.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with limits. This problem says, assess the validity of the following statement and provide a reasoning. Assume D and P are finite numbers. If the limit as Z approaches D of K of Z is equal to P, then K of D is equal to P, we're given four possible choices as our answers. For choice a true because the function value at D must match the limit of the function as Z approaches D for choice B false. Because the limit of the function as Z approaches D does not necessarily imply the function's value at D is equal to the limit for choice C we have true because the function is continuous at D. And for choice D, we have false because the limit cannot be determined. So we need to determine in this um problem whether the statement they were given in the problem is true or false. And so the statement that we're looking at is if the limit of Z approaching D of K of Z is equal to P, then K of D is equal to P and we're gonna focus on the first part of the statement dealing with the limit. So that first part, we have the limit as Z approaches D of K of Z is equal to P. And so recall that when you're taking a limit of a function, what that means is that as Z approaches D, then K of Z approaches P and note that in that definition, it doesn't say that K of D has to equal P. It just says that K of Z has to approach P as Z approaches D. So let's see if we can come up with a counterexample to this statement. And for this counterexample, we're gonna use the function K of Z is equal to the quantity of Z squared minus one in quantity divided by the quantity of Z minus one. And so what we're going to do is take the limit as Z approaches one of KFZ. And when we take this limit, this limit is going to be equal to two. The way you could find this limit is to factor the numerator. And you'll get, when you factor that numerator, you'll get Z minus one multiplied by Z plus one, the Z minus ones will cancel out leaving Z plus one. And then setting um Z equal to one for our limit, we get a limit of two. However, if we look at the value of the function at Z equal to one and so we'll have K of one and that's gonna be equal to zero, divided by zero, which is undefined since I'm dividing by zero. So here we've shown an example where we have taken the limit and gotten a value. And when we substituted in Z with uh the value that we are taking the limit of into our function, we actually get a different value in this case, an undefined value. So that means our statement here is false. And that's because the limit of the function as Z approaches D does not necessarily necessarily imply that the functions value at D is equal to the limit. And so if we look at our answer choices, turns out that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been helpful and I'll see you in the next video.

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.
If limx→a f(x) = L, then f(a)=L.

Key Concepts

Limit of a Function

Continuity

Counterexample

Watch next