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• - [Instructor] We are asked to find

• these three different limits.

• I encourage you like always,

• pause this video and try to do it yourself

• before we do it together.

• So when you do this first one,

• you might just try to find the limit

• as x approaches negative two of f of x

• and then the limit as x approaches negative two of g of x

• and then add those two limits together.

• But you will quickly find a problem,

• 'cause when you find the limit

• as x approaches negative two of f of x,

• it looks as we are approaching negative two

• from the left, it looks like we're approaching one.

• As we approach x equals negative two from the right,

• it looks like we're approaching three.

• So it looks like the limit

• as x approaches negative two of f of x doesn't exist,

• and the same thing's true of g of x.

• If we approach from the left,

• it looks like we're approaching three.

• If we approach from the right,

• it looks like we're approaching one.

• But it turns out that this limit can still exist

• as long as the limit as x approaches negative two

• from the left of the sum,

• f of x plus g of x,

• exists and is equal to the limit

• as x approaches negative two from the right of the sum,

• f of x plus g of x.

• So what are these things?

• Well, as we approach negative two from the left,

• f of x is approaching, looks like one,

• and g of x is approaching three.

• So it looks like we're approaching one and three.

• So it looks like this is approaching.

• The sum is going to approach four.

• And if we're coming from the right,

• f of x looks like it's approaching three

• and g of x looks like it is approaching one.

• Once again, this is equal to four.

• And since the left and right handed limits

• are approaching the same thing,

• we would say that this limit exists and it is equal to four.

• Now let's do this next example as x approaches one.

• Well, we'll do the exact same exercise.

• And once again, if you look at the individual limits

• for f of x from the left and the right as we approach one,

• this limit doesn't exist.

• But the limit as x approaches one of the sum might exist,

• so let's try that out.

• So the limit as x approaches one

• from the left hand side of f of x plus g of x,

• what is that going to be equal to?

• So f of x, as we approach one from the left,

• looks like this is approaching two.

• I'm just doing this for shorthand.

• And g of x, as we approach one from the left,

• it looks like it is approaching zero.

• So this will be approaching two plus zero, which is two.

• And then the limit,

• as x approaches one from the right hand side

• of f of x plus g of x is going to be equal to.

• Well, for f of x as we're approaching one

• from the right hand side,

• looks like it's approaching negative one.

• And for g of x as we're approaching one

• from the the right hand side,

• looks like we're approaching zero again.

• Here it looks like we're approaching negative one.

• So the left and right hand limits

• aren't approaching the same value,

• so this one does not exist.

• And then last but not least,

• x approaches one of f of x times g of x.

• So we'll do the same drill.

• Limit as x approaches one from the left hand side

• of f of x times g of x.

• Well, here, and we can even use the values here.

• We see it was approaching one from the left.

• We are approaching two, so this is two.

• And when we're approaching one from the left here,

• we're approaching zero.

• We're gonna be approaching two times zero, which is zero.

• And then we approach from the right.

• X approaches one from the right

• of f of x times g of x.

• Well, we already saw when we're approaching one

• from the right of f of x,

• we're approaching negative one.

• But g of x, approaching one from the right,

• is still approaching zero,

• so this is going to be zero again, so this limit exists.

• We get the same limit

• when we approach from the left and the right.

• It is equal to zero.

• So these are pretty interesting examples,

• because sometimes when you think

• that the component limits don't exist

• that that means that the sum

• or the product might not exist,

• but this shows at least two examples

• where that is not the case.

- [Instructor] We are asked to find

A2 初級 美國腔

# 1.43分段函數（Limits of combined functions: piecewise functions | AP Calculus AB | Khan Academy）

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yukang920108 發佈於 2022 年 07 月 01 日