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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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# Limits from graphs | Limits and continuity | AP Calculus AB | Khan Academy

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yukang920108 posted on 2022/06/28
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