Video: Finding the Limit of a Product of Two Functions

Assume that lim_(π‘₯ β†’ 2) 𝑓(π‘₯) = 5. Find lim_(π‘₯ β†’ 2) (π‘₯ + 2) β‹… 𝑓(x).

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Video Transcript

Assume that the limit as π‘₯ approaches two of 𝑓 of π‘₯ is equal to five. Find the limit as π‘₯ approaches two of π‘₯ plus two times 𝑓 of π‘₯.

The question tells us for some function 𝑓 of π‘₯, the limit as π‘₯ approaches two of 𝑓 of π‘₯ is equal to five. We need to use this information to find the limit as π‘₯ approaches two of π‘₯ plus two times 𝑓 of π‘₯. We don’t know how to directly evaluate this limit, so we’ll need to rewrite it in terms of limits we do know how to evaluate.

We know how to evaluate the limit of π‘₯ plus two. And we know how to evaluate the limit of 𝑓 of π‘₯ as π‘₯ approaches two. So, we want to rewrite the limit of a product as the product of the limits. And luckily, we know that we can do this. We know for any functions 𝑔 and β„Ž and any real constant π‘Ž, the limit as π‘₯ approaches π‘Ž of 𝑔 of π‘₯ times β„Ž of π‘₯ is equal to the limit as π‘₯ approaches π‘Ž of 𝑔 of π‘₯ times the limit as π‘₯ approaches π‘Ž of β„Ž of π‘₯. In other words, the limit of the product of two functions is equal to the product of the limit of those two functions.

To apply this to the limit given to us in the question, we’ll set our value of π‘Ž equal to two, our function 𝑔 of π‘₯ to be π‘₯ plus two, and our function β„Ž of π‘₯ to be 𝑓 of π‘₯. So, by using these values, we have the limit as π‘₯ approaches two of π‘₯ plus two times 𝑓 of π‘₯ is equal to the limit as π‘₯ approaches two of π‘₯ plus two multiplied by the limit as π‘₯ approaches two of 𝑓 of π‘₯.

We can now evaluate both of these limits. We see the limit of π‘₯ plus two is the limit of a linear function. We can evaluate this by direct substitution. So, we substitute π‘₯ is equal to two into our linear function π‘₯ plus two. This just gives us two plus two, which of course simplifies to give us four. And of course, we’re told the limit as π‘₯ approaches two of 𝑓 of π‘₯ is equal to five. So, we can evaluate this limit to just give us five. So, this gives us four times five, which is just equal to 20.

So, we’ve shown if the limit as π‘₯ approaches two of some function 𝑓 of π‘₯ is equal to five, then the limit as π‘₯ approaches two of π‘₯ plus two times 𝑓 of π‘₯ must be equal to 20.

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