# Limit of the function

#### wishmaster

##### Active member
I know i asked similar questions multiple times,but again i have a problem,seems im not good with roots.....
I have to calculate the following limit:

$$\displaystyle \lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$$

#### MarkFL

Staff member
What do you get if you try substituting $x=0$ into the expression?

#### wishmaster

##### Active member
What do you get if you try substituting $x=0$ into the expression?
I get $0$ of course...so i know i should do some operations,so i get rid of the x in the denumerator.

#### MarkFL

Staff member
I get $0$ of course...so i know i should do some operations,so i get rid of the x in the denumerator.
You actually get $$\displaystyle \frac{0}{0}$$, and this is a dreaded indeterminate form. So, since you have seen problems like this before, what would you say we need to do to get it into a determinate form? What would be a good strategy?

#### wishmaster

##### Active member
You actually get $$\displaystyle \frac{0}{0}$$, and this is a dreaded indeterminate form. So, since you have seen problems like this before, what would you say we need to do to get it into a determinate form? What would be a good strategy?
maybe to move the roots somehow into the denumerator?

#### MarkFL

Staff member
maybe to move the roots somehow into the denumerator?
Correct, and how can we accomplish this?

#### wishmaster

##### Active member
Correct, and how can we accomplish this?
To multiply with $$\displaystyle \sqrt{1+x}-\sqrt{1-x}$$ ?

#### MarkFL

Staff member
To multiply with $$\displaystyle \sqrt{1+x}-\sqrt{1-x}$$ ?
No, you want to use the conjugate of the numerator, this way the radicals will disappear from the numerator.

#### wishmaster

##### Active member
No, you want to use the conjugate of the numerator, this way the radicals will disappear from the numerator.
How?

#### MarkFL

Staff member
What is the conjugate of the numerator?

#### wishmaster

##### Active member
What is the conjugate of the numerator?
$$\displaystyle 1-x$$ ??
Im stupid it seems.....

#### MarkFL

Staff member
$$\displaystyle 1-x$$ ??
Im stupid it seems.....
Hey, don't get discouraged...you are learning...it takes time.

Consider the expression $a+b$. It's conjugate is $a-b$. Do you see that if we multiply them together, we will have a difference of squares, and a squared radical is no loger a radical. So now what would you say the conjugate of the numerator is?

#### wishmaster

##### Active member
Hey, don't get discouraged...you are learning...it takes time.

Consider the expression $a+b$. It's conjugate is $a-b$. Do you see that if we multiply them together, we will have a difference of squares, and a squared radical is no loger a radical. So now what would you say the conjugate of the numerator is?
$$\displaystyle \sqrt{1+x}+\sqrt{1-x}$$ ?

#### MarkFL

Staff member
$$\displaystyle \sqrt{1+x}+\sqrt{1-x}$$ ?
Yes, good!

Now, you want to multiply the expression for which you are asked to find the limit by $1$ in the form of this conjugate divided by itself:

$$\displaystyle 1=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$$

What do you find?

#### wishmaster

##### Active member
Yes, good!

Now, you want to multiply the expression for which you are asked to find the limit by $1$ in the form of this conjugate divided by itself:

$$\displaystyle 1=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$$

What do you find?
But how turned the denumerator into this term? I had $x$ in it.

#### MarkFL

Staff member
But how turned the denumerator into this term? I had $x$ in it.
This is what you want to multiply the expression with. Since it is $1$, you aren't changing its value. So you want:

$$\displaystyle \lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}\cdot\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$$

#### wishmaster

##### Active member
This is what you want to multiply the expression with. Since it is $1$, you aren't changing its value. So you want:

$$\displaystyle \lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}\cdot\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$$
So i multiply all together?

#### MarkFL

Staff member
So i multiply all together?
Yes. You will find you will be able to get rid of the $x$ in the denominator as well when you reduce.

#### wishmaster

##### Active member
Yes. You will find you will be able to get rid of the $x$ in the denominator as well when you reduce.

$$\displaystyle \frac{2x}{x(\sqrt{1+x}+\sqrt{1-x})}$$ ?

#### Petrus

##### Well-known member
I know i asked similar questions multiple times,but again i have a problem,seems im not good with roots.....
I have to calculate the following limit:

$$\displaystyle \lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$$
Hello wishmaster!
I bet it's mather of time until you learn l'hopitals rule which i would use when I see this limit here you got a Link that explain it well Pauls Online Notes : Calculus I - L'Hospital's Rule and Indeterminate Forms
$$\displaystyle |\pi\rangle$$
$$\displaystyle \frac{2x}{x(\sqrt{1+x}+\sqrt{1-x})}$$ ?