Main content

## Calculus 1

### Course: Calculus 1 > Unit 1

Lesson 2: Estimating limits from graphs- Estimating limit values from graphs
- Unbounded limits
- Estimating limit values from graphs
- Estimating limit values from graphs
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- One-sided limits from graphs
- Connecting limits and graphical behavior
- Connecting limits and graphical behavior
- Connecting limits and graphical behavior (more examples)

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Estimating limit values from graphs

Worked examples of estimating limits of a function from its graph.

## Want to join the conversation?

- At5:47, why can't we say that the limit is infinity?(89 votes)
- "In order for a limit to exist, the function has to approach a particular value. In the case shown above, the arrows on the function indicate that the the function becomes infinitely large. Since the function doesn't approach a particular value, the limit does not exist."(125 votes)

- Does there have to be a gap in the equation to find the limit? What's the purpose of this? Also, when we find this number, is the number itself called the equation's "limit"?(10 votes)
- No, there does not have to be a gap. They are just for you understand that even if there is a "gap", by which I mean the function at that particular value is undefined or defined as something else that does not match the kind of approaching behavior of the function, the limit is not affected. The 'number itself' will not be the equation's limit, as it only refers to the limit of the function as x approaches one certain point on it.(21 votes)

- Hello, how come on the second graph, the Lim of g(7) is 2? Wouldn't the dot make the limit become two values, therefore making it not exist? I'm confused. Thank you!(12 votes)
- At g(7), the limit would not have two values. This is because the filled in dot is where the line is. The unfilled dot is where the two lines converge, but it is not a point on the line. You can think of it like a jump graph.(8 votes)

- In the first graph, why does the limit as x approaches 2 not exist? Why can't we define limit as two values.. does that not mean that the function of the given parabola is wrong? Should they not be two different functions?(6 votes)
- We cannot define limit as two values. You'll learn about the derivatives and you'll know that it's impossible to have two values. You'll have to find the slope of the tangent line at the certain point of x. And if the left limit and the right limit is different, there will be two slopes which are not possible. The slope could be also defined as a velocity. You cannot have two velocities at the same time. It's hard to explain if you didn't learn derivatives.

Hope this helps! If you have any questions or need help, please ask! :)(17 votes)

- Why is it called limit ? i have no idea why they call it limit. it does not seem to relate what it means(7 votes)
- "Limit" simply relates to the fact that the distance between x and c is gradually "limited" to zero.(8 votes)

- What's the difference between a function and a limit?(3 votes)
- A function is a relation between a set of inputs and permissible outputs with the property that each input is related to exactly one output. A limit is the value that a function approaches as the input approaches a given value.(14 votes)

- At5:56why doesn't the limit exist? I thought it would be +infinity(4 votes)
- Positive infinity is not an "existent" limit. It is undefined or nonexistent. Alternatively, you could say that the limit tends off to infinity.(5 votes)

- At2:45it is stated that it does not exist because there are two different values, but in a similar case at5:02there is a value. Could I get some clarification on this?(4 votes)
- At2:45as x approaches 2 the lines on the graph are heading toward different values. The line on the left side is heading towards 2 while the line on the right side is heading towards 5. Since the lines are heading toward different values as x->2, the limit does not exist. However, at5:02, the lines ARE heading towards the same value, even though the function is not necessarily defined at the point they are heading towards.(4 votes)

- for the second graph, wouldn't the limit of G(x) as it approaches 7 equal DNE? it because it has a point directly above the limit?(3 votes)
- This limit as x approaches 7 DOES exist and has a y value of 2. g(7) is the actual value when x=7 and therefore is a closed circle. g(7) is different than the value of the limit when approaching 7. For the limit to NOT exist, you would need two seperate lines approaching two different values.(5 votes)

- At4:50, if we take a value infinitely close to 7 like 6..999999999999, then this value would be almost 7 therefore would give us the value of g(x) as 5, right? How can the value of g(x) be equal to 2?(3 votes)
- The value of g(x) is
*not*2 anywhere in that region. We can make the output of g(x) as close to 2 as we like by picking values of x as close to 7 as we like.

If you meant 6.999999999999 to be a 6 followed by twelve 9's, that number is not infinitely close to 7, it differs from 7 by 10^(-12). Inputting this number gives an output very close to, but not equal, to 2.

6.9... with 9's continuing indefinitely is equal to 7, precisely. 6.9... and 7 are different names for the same number. Putting this value in gives us 5 because g(7)=5.(4 votes)

## Video transcript

- [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.