情態助動詞 A2

- [Voiceover] Let's see if we can find the limit

over the square root of x plus 5 minus 2.

okay, well let's just use our limit properties

this is going to be the same thing as the limit

if we think about the graph y equals x plus 1,

it's continuous everywhere, especially at

x equals negative 1, and so to evaluate this limit,

we just have to evaluate this expression at

so this numerator's just going to evaluate to negative 1

square root of x plus 5 minus 2 isn't continuous everywhere

but it is continuous at x equals negative 1

so this is going to be the square root of negative 1

and in the denominator, negative 1 plus 5 is 4,

Now, when you see that, you might be tempted to give up.

You say, oh, look, there's a zero in the denominator,

And if this was non-zero up here in the numerator,

if you're taking a non-zero value and dividing it by zero,

it doesn't mean necessarily that your limit does not exist,

and as we'll see in this video and many future ones,

there are tools at our disposal to address this,

Now, the tool that we're going to look at

is is there another way of rewriting this expression

so let's take this thing right over here,

so essentially what we're trying to do is find the limit

x plus 1 and the only reason I'm defining it as g of x

is just to be able to think of it more clearly as a function

or x plus 1 over the square root of x plus 5 minus 2.

Now, the technique we're going to use is when you get

this indeterminate form and if you have a square root

in either the numerator or the denominator,

it might help to get rid of that square root

and this is often called rationalizing the expression.

In this case, you have a square root in the denominator,

is we would be leveraging our knowledge of difference

and we were to multiply that times the square root of a

minus b, well that would be the square root of a squared

is we're going to multiply the numerator and the denominator

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