In a general sense limits allow us to determine what value a function is approaching when
we use a particular input.
Not necessarily what the function gives us as output, but rather what value its getting
arbitrarily close to.
Let's explain limits using an analogy.
Suppose you are watching T.V. and start getting a massive craving for pizza.
Fortunately you just happen to have some left over pizza in the kitchen.
So you get up from the couch and start heading towards the fridge.
In this instance if someone were to describe where you are going, they would say you are
approaching the fridge.
They would be confident in this description because as you keep walking, you are getting
closer and closer to where the fidge is located.
This example is the same thing we want to do with functions.
When we take the limit of a function we are describing where they are going!
Let's see an example of this with the function f(x) = 3x^2 - 1
For this function I'm really curious what value the function is approaching as I use
x values close to the number 2.
Let's see this by using some inputs 1.9, 1.99, and 1.999.
When I use these, I get values such as 9.83, 10.8803, and 10.988003.
From these values is appears that the function is approaching 11.
So we say the limit of the function as x approaches 2, is 11.
Remember What I'm really saying here is that we can get arbitrarily close to the number
11, I just have to pick values that are sufficently close to 2 in order to do it.
Now at this point you might be thinking, that's fantastic, but couldn't you have found the
limit simplying by plugging 2 into the function.
Wouldn't that also give you 11?
In this instance the answer is yes, but the focus with a limit should be on what value
its approaching, and there are some functions where you simply can't plug in a number to
find the limit.
In otherwords, they are not always the same.
Let's cover this by going back to our pizza analogy.
Like before you have been struck with a craving for pizza so you are headed toward the fidge
for a quick snack.
Now in one scenario the fridge is there, loaded with pizza, and you can easily satify your
craving for pepperoni.
But in an alternate scenario the fridge is gone, possible stolen by pizza craving ninjas,
and you are left empty handed.
Even though both of these situations are completely different, your behavior leading up to them
is exactly the same.
In either case you were still approaching the fridge.
This is the key difference with limits, they are used to describe what value a function
is approaching.
They are not used to describe the value the function actually reaches.
Let's see how this works with yet another function.
Let's go ahead and use (x^2 - 4) / x-2 Like before we are interested in what value
the function approaches we use x values close to 2.
Let's go ahead and choose some inputs like 1.9, 1.99, and 1.999.
When we use these we get the values of 3.9, 3.99, and 3.999.
From these it appears that the function is approaching 4.
So again we say the limit of the function as x approaches 2, is 4.
If you try and find this value by instead plugging in 2, something strange happens.
When you plug 2 into the function, you get zero divided by zero.
This shows that the function doesn't actually ever get to 4.
In fact a quick look at the graph shows a hole right at 4.
Despite the hole, the behavior of the function leading up to it is the same.
Since the behavior is the same, we still say that the limit of the function as x approaches
2 is 4, even though it never actually gets there.
Hopefully both of these examples really highlight how limits focus on the behavior of a function,
what they get arbitrarily close to.
One thing we still have left to cover is what it exactly means when we say a function gets
"arbitrarily close" to a value.
But don't worry, We'll be able to tackle that tricky problem in the next video when we introduce
epsilon and delta.
Thanks for watching.
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If you want to know more about limits, you can watch a few examples here.
You can also move onto my next lecture video where I talk about the precise definition
of a limit using epsilon and delta!
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