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

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

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True or False: lim_(π‘₯ β†’ π‘Ž) 𝑓(π‘₯) = 𝑙 means that as π‘₯ gets closer to π‘Ž, 𝑓(π‘₯) gets closer to 𝑙.

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Video Transcript

True or False: The limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ is equal to 𝑙 means that as π‘₯ gets closer to π‘Ž, 𝑓 of π‘₯ gets closer to 𝑙.

In this question, we need to determine whether a statement is true or false. We need to determine whether saying that the limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ is equal to 𝑙 means that as π‘₯ gets closer to π‘Ž, 𝑓 of π‘₯ gets closer to 𝑙. And to answer this question, let’s start by recalling what we mean by this limit notation. We can recall that we say if the values of 𝑓 of π‘₯ approach some finite value of 𝑙 as the values of π‘₯ approach π‘Ž from either side, but not necessarily when π‘₯ is equal to π‘Ž, then we say that the limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ is equal to 𝑙. This is the definition of what this limit notation means. And we can rewrite and reorder this information slightly.

For example, saying that π‘₯ approaches π‘Ž from either side but not necessarily when π‘₯ is equal to π‘Ž is exactly the same as saying our values of π‘₯ are getting closer and closer to π‘Ž. Similarly, saying that when this happens the values of 𝑓 of π‘₯ are approaching some finite value of 𝑙 is exactly the same as saying our values of 𝑓 of π‘₯ are getting closer and closer to this value of 𝑙. So this statement is exactly the same as our definition; we’ve just reordered the information slightly.

Therefore, we can say it’s true that saying the limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ is equal to 𝑙 means that as π‘₯ gets closer to π‘Ž, 𝑓 of π‘₯ gets closer to 𝑙.

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