Find the limit as x approaches zero from the right of ((sqrt((1+x)/(x^2))-(1/x))

Question

Find the limit as x approaches zero from the right of ((sqrt((1+x)/(x^2))-(1/x))

I don't even know where to start with this. Help?

P.S. If you go to the wolfphram alpha website and put copy and paste ((sqrt((1+x)/(x^2))-(1/x)) into the search bar you can see what it looks like.

Thanks!

Answer 1

as long as you're at wolframalpha, type in

limit x->0+ (sqrt((1+x)/(x^2))-(1/x))

and see that the limit is 1/2. The question is, how do you figure it?

√((1+x)/x^2) = √(1+x)/x
so,

√((1+x)/(x^2))-(1/x) = √(1+x)/x - 1/x
= (√(1+x) - 1)/x

As x->0, the fraction is 0/0, so use L'Hospital's Rule to get

1/(2√(1+x)) / 1 = 1/2

Answer 2

Oh, I forgot all about L'Hospital's Rule. Thanks!

Answer 3

To find the limit as x approaches zero from the right of the given expression:

((sqrt((1+x)/(x^2))-(1/x))

We need to simplify the expression first.

Start by simplifying the square root part:

(sqrt((1+x)/(x^2)))

To write it in a simpler form, you can multiply the numerator and denominator by sqrt(x^2):

= sqrt((1+x) * sqrt(x^2))/(x^2 * sqrt(x^2))

= sqrt((1+x) * sqrt(x^2))/(x^3)

Now, let's simplify the whole expression:

((sqrt((1+x)/(x^2))-(1/x))

= sqrt((1+x) * sqrt(x^2))/(x^3) - (1/x)

To take the limit as x approaches zero from the right, we substitute x = 0 into the expression:

sqrt((1+0) * sqrt(0^2))/(0^3) - (1/0)

Notice that the term (1/x) becomes undefined as x approaches zero from the right, as we cannot divide by zero.

Therefore, the limit of the given expression as x approaches zero from the right is also undefined.

If you still want to verify this using Wolfram Alpha, you can follow the steps below:

1. Open the Wolfram Alpha website (www.wolframalpha.com).
2. Copy and paste the expression ((sqrt((1+x)/(x^2))-(1/x)) into the search bar.
3. Hit enter or click the "equal" button.
4. Wolfram Alpha will display the result, along with additional information and a graphical representation of the expression.

Keep in mind that while Wolfram Alpha can solve complex mathematical problems and display results, it's always important to understand the underlying concepts and steps involved in solving them.