Question Video: Understanding the Limit of a Function at a Point | Nagwa Question Video: Understanding the Limit of a Function at a Point | Nagwa

# Question Video: Understanding the Limit of a Function at a Point Mathematics • Second Year of Secondary School

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Which of the following statements is not the same as saying that lim_(π₯ β 8) π(π₯) = 3? [A] We can make π(π₯) as close as we like to 3 by taking π₯ sufficiently close to 8. [B] π(3) is equal to π(8). [C] As π₯ gets closer and closer to 8, π(π₯) gets closer and closer to 3. [D] π(π₯) approaches 3 as π₯ approaches 8.

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### Video Transcript

Which of the following statements is not the same as saying that the limit as π₯ approaches eight of π of π₯ is equal to three? Is it option (A) we can make π of π₯ as close as we like to three by taking π₯ sufficiently close to eight? Is it option (B) π evaluated at three is equal to π evaluated at eight? Option (C) as π₯ gets closer and closer to eight, π of π₯ gets closer and closer to three. Or is it option (D) π of π₯ approaches three as π₯ approaches eight?

In this question, weβre given four statements, and we need to determine which of these four statements is not the same as saying that the limit as π₯ approaches eight of π of π₯ is equal to three. To answer this question, letβs start by recalling what we mean by the value of the limit of a function at a point. We recall that we say if the values of π of π₯ approach some finite value of πΏ, as the values of π₯ approach some value of π from either side but not necessarily when π₯ is equal to π, then we say that the limit as π₯ approaches π of π of π₯ is equal to πΏ. And we can directly use this definition to answer the question. However, letβs start by replacing the values of π and πΏ in the definition with the values given in the question.

Weβre taking the limit as π₯ approaches eight of our function π of π₯, so weβll set our value of π equal to eight. And we say that this limit is equal to three, so weβll set our value of πΏ equal to three. We can then update our definition. This now tells us since the limit as π₯ approaches eight of π of π₯ is equal to three as the values of π₯ approach eight from either side but not necessarily when π₯ is equal to eight, then the values of π of π₯ must be approaching three. And this is really useful because we can see that this statement is equivalent to three of our options.

First, we can make π of π₯ as close as we like to three by taking π₯ sufficiently close to eight. This is a direct result from our definition. Our values of π of π₯ are approaching three as our values of π₯ approach eight. And since π of π₯ approaches three, we can make this as close as we like by taking π₯ sufficiently close to eight. So, option (A) is exactly the same as saying the limit as π₯ approaches eight of π of π₯ is equal to three.

We can see the same is true in option (C). This is exactly the same as saying the limit as π₯ approaches eight of π of π₯ is equal to three. Itβs almost exactly the same as what is written. We only need to note one thing. When we say that our values of π₯ are getting closer and closer to eight, we do mean that π₯ can approach from either side and we donβt need to know what happens when π₯ is equal to eight. And this is exactly the same as our definition. As our values of π₯ approach eight from either side but not necessarily when π₯ is equal to eight, the values of π of π₯ are approaching three.

Finally, in option (D), we see that π of π₯ approaches three as π₯ approaches eight. And once again, this is exactly the same as what is written. So, this option is also the same as saying the limit as π₯ approaches eight of π of π₯ is equal to three. Now, this is enough to answer our question. The other three options are all the same. So, option (B) must be different. However, for due diligence, letβs show this.

This statement says that the value of π of three needs to be equal to π evaluated at eight. Of course, we can immediately notice a few things wrong. For example, this statement does not tell us any information about the values of π of π₯ as our values of π₯ get closer and closer to eight. And we can also notice in our definition we donβt need to know the value of our function when π₯ is equal to eight. The exact value of π evaluated at eight will have no effect on the limit as π₯ approaches eight of π of π₯. In particular, our function does not even need to be defined at π₯ is equal to eight.

So, knowing that the limit as π₯ approaches eight of π of π₯ is equal to three wonβt tell us any information about the function evaluated at eight or the function evaluated at three. So, we can say that option (B) saying that π evaluated at three is equal to π evaluated at eight is not the same as saying that the limit as π₯ approaches eight of π of π₯ is equal to three.

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