Video: Finding the Sum of a Complex Number and Its Conjugate

What is the correct notation that describes the following statement? As π‘₯ approaches 0, 𝑓(π‘₯) approaches βˆ’6.

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Video Transcript

What is the correct notation that describes the following statement? As π‘₯ approaches zero, 𝑓 of π‘₯ approaches negative six.

For a question of this type, the first thing we might notice is this word approaches. Whenever we are told that the value of a function or a variable approaches something, it gives us a hint that our question might involve limits. The standard notation for a limit is shown here. And we would read the statement as: the limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ is equal to 𝐿. Breaking it down, what this statement is telling us is that as the value of π‘₯ approaches the constant, here called π‘Ž, the value of 𝑓 of π‘₯ will approach 𝐿. This 𝐿 is sometimes referred to as the value of the limit. A quick tip here is to remember that we’re not being told that 𝑓 of π‘₯ is equal to 𝐿, but rather that the value of our limit is equal to 𝐿.

Okay, now that we understand how to write a limit and how to interpret the statement, let’s see how it applies to our question. We are told that as π‘₯ approaches zero, 𝑓 of π‘₯ approaches negative six. This zero is the value which we consider π‘₯ to be approaching. And it’s represented by π‘Ž in our general limit equation. Similarly, this negative six, which is the value 𝑓 of π‘₯ approaches, is represented by 𝐿 in our general limit equation. We therefore write the following statement. The limit as π‘₯ approaches zero of 𝑓 of π‘₯ is equal to negative six. Since we have substituted π‘Ž is zero and 𝐿 is negative six, the statement is interpreted in the following way. As the value of π‘₯ approaches zero, the value of 𝑓 of π‘₯ approaches negative six.

Looking at the statement in the question and the statement that we have just written, these exactly match. This means that we have expressed the statement given in the question using the correct notation for a limit.

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