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  • - [Instructor] So we have the graph of Y equals f of x

  • right over here and we want to figure out

  • three different limits and like always

  • pause this video and see if you can figure it out

  • on your own before we do it together.

  • Alright now first let's think about what's the limit

  • of f of x it's x approaches six.

  • So as x, I'm gonna do this in a color you can see,

  • as x approaches six from both sides

  • well as we approach six from the left hand side,

  • from values less than six,

  • it looks like our f of x is approaching one

  • and as we approach x equals six from the right hand side

  • it looks like our f of x is once again approaching one

  • and in order for this limit to exist,

  • we need to be approaching the same value

  • from both the left and the right hand side

  • and so here at least graphically,

  • so you never are sure with a graph but this is

  • a pretty good estimate, it looks like we are approaching one

  • right over there, in a darker color.

  • Now let's do this next one.

  • The limit of f of x is x approaches four

  • so as we approach four from the left hand side

  • what is going on?

  • Well as we approach four from the left hand side

  • it looks like our function, the value of our function

  • it looks like it is approaching three.

  • Remember you can have a limit exist at an x value

  • where the function itself is not defined,

  • the function , if you said after four, it's not defined

  • but it looks like when we approach it from the left

  • when we approach x equals four from the left

  • it looks like f is approaching three

  • and then we approach four from the right,

  • once again, it looks like our function is approaching three

  • so here I would say, at least from what we can tell

  • from the graph it looks like the limit

  • of f of x is x approaches four is three,

  • even though the function itself is not defined yet.

  • Now let's think about the limit as x approaches two.

  • So this is interesting the function is defined there

  • f of two is two, let's see when we approach

  • from the left hand side it looks like our function

  • is approaching the value of two

  • but when we approach from the right hand side,

  • when we approach x equals two from the right hand side,

  • our function is getting closer and closer to five

  • it's not quite getting to five but as we go from

  • you know 2.1 2.01 2.001 it looks like our function

  • the value of our function's getting closer and closer

  • to five and since we are approaching two different values

  • from the left hand side and the right hand side

  • as x approached two from the left hand side

  • and the right hand side we would say that this limit

  • does not exist so does not exist.

  • Which is interesting.

  • In this first case the function is defined at six

  • and the limit is equal to the value of the function

  • at x equals six, here the function was not defined

  • at x equals four, but the limit does exist

  • here the function is defined at f equals, x equals two

  • but the limit does not exist as we approach x equals two

  • let's do another function just to get more cases

  • of looking at graphical limits.

  • So here we have the graph of Y is equal to g of x

  • and once again pause this video and have a go at it

  • and see if you can figure out these limits graphically.

  • So first we have the limit as x approaches five

  • g of x so as we approach five from the left hand side

  • it looks like we are approaching this value

  • let me just draw a straight Line that takes us

  • so it looks like we're approaching this value

  • and as we approach five from the right hand side

  • it also looks like we are approaching that same value.

  • And so this value, just eye balling it off of here

  • looks like it's about .4 so I'll say this limit

  • definitely exists just when looking at a graph

  • it's not that precise

  • so I would say it's approximately 0.4

  • it might be 0.41 it might be 0.41456789

  • we don't know exactly just looking at this graph

  • but it looks like a value roughly around there.

  • Now let's think about the limit of g of x

  • as x approaches seven so let's do the same exercise.

  • What happens as we approach from the left

  • from values less than seven 6.9, 6.99, 6.999

  • well it looks like the value of our function

  • is approaching two, it doesn't matter

  • that the actual function is defined g of seven is five

  • but as we approach from the left,

  • as x goes 6.9, 6.99 and so on,

  • it looks like our value of our function

  • is approaching two, and as we approach x equals seven

  • from the right hand side it seems like the same thing

  • is happening it seems like we are approaching two

  • and so I would say that this is going to be equal to two

  • and so once again, the function is defined there

  • and the limit exists there but the g of seven

  • is different than the value if the limit of g of x

  • as x approaches seven.

  • Now let's do one more.

  • What's the limit as x approaches one.

  • Well we'll do the same thing,

  • from the left hand side, it looks like we're going

  • unbounded as x goes .9, 0.99, 0.999 and 0.9999

  • it looks like we're just going unbounded towards infinity

  • and as we approach from the right hand side

  • it looks like the same thing is happening

  • we're going unbounded to infinity.

  • So formally, sometimes informally people will say

  • oh it's approaching infinity or something like that

  • but if we wanna be formal about what a limit means

  • in this context because it is unbounded

  • we would say that it does not exist.

  • Does not exist.

- [Instructor] So we have the graph of Y equals f of x

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圖表的限制 |限制和連續性 | AP微積分AB |可汗學院(Limits from graphs | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 06 月 28 日
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