Name: One-sided limits from graphs | Limits | Differential Calculus | Khan Academy
Uploaded: 2022-06-30T02:49:56.000Z
Duration: 9 min 11 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

To-infinitive

is the value of our function approaching--

as we approach x equals 2 from values less than x equals 2.

So as you imagine, as we approach x equals 2--

So x equals 1, x equals 1.5, x equals 1.9, x equals 1.999,

And we see that f of x seems to be approaching

and we're going to specify the direction-- as x approaches 2

the negative as a superscript after the 2

to denote the direction that we're approaching.

We're approaching 2 from the negative direction.

We're approaching 2 from values less than 2.

We're getting closer and closer to 2, but from below-- 1.9,

As x gets closer and closer from those values,

And we see here that it is approaching 5.

But what if we were asked the natural other question-- What

So this is a little superscript positive right over here.

So now we're going to approach x equals 2,

but we're going to approach it from this direction--

x equals 3, x equals 2.5, x equals 2.1, x equals 2.01,

And we're going to get closer and closer to 2,

but we're coming from values that are larger than 2.

So here, when x equals 3, f of x is here.

When x equals 2.01, f of x looks like it's right over here.

So in this situation, we're getting closer and closer

It actually then just has a jump discontinuity.

This seems to be the limiting value when we approach when

we approach 2 from values greater than 2.

And so when we think about limits in general,

the only way that a limit at 2 will actually exist

And as we approach 2 from values above 2,

So in this case, the limit-- let me write this down--

equal the limit of f of x, as x approaches 2

And since this is the case-- that they're not equal--

of f of x-- so the limit of f of x, as x approaches 2,

In order for it to have existed, these two things

would have had to have been equal to each other.

what is the limit of f of x as x approaches 4?

Well, then we could think about the two one-sided limits--

the one-sided limit from below and the one-sided limit

The limit of f of x, as x approaches 4 from below-- so

So when x equals 3, we're here where f of 3

is negative 2. f of 3.5 seems to be right over here.

f of 3.9 seems to be right over here. f of 3.999--

we're getting closer and closer to our function equaling

So the limit as we approach 4 from below--

we could say-- this is going to be equal to negative 5.

And if we were to ask ourselves the limit of f

from values larger than 4, well, same exercise.

f of 4.1 seems right about here. f of 4.01

but we're getting closer and closer to it.

And we see, once again, we are approaching 5.

Even if f of 4 was not defined on either side,

And since the limit from the left-hand side

is equal to the limit from the right-hand side,

we can say-- so these two things are equal.

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One-sided limits from graphs | Limits | Differential Calculus | Khan Academy

approach

figure

situation

positive