Question

Find the value of limit `{sqrt(1+x^2)-sqrt(1+x)}/{sqrt(1+x^3)-sqrt(1+x)}` as x approaches to 0?

Answer 1

`1`

Explanation:

`{(sqrt(1+x^2)-sqrt(1+x))}/{(sqrt(1+x^3)-sqrt(1+x))}`

Remove the subtraction of radicals by multiplying numerator and denominator by

`(sqrt(1+x^2) + sqrt(1+x))(sqrt(1+x^3) + sqrt(1+x))`.

(That is the product of the conjugates of the numerator and the denominator.)

We get

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

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

Evaluating the limit as `xrarr0`, we get

`= (sqrt(1+0)+sqrt(1+0))/((0+1)(sqrt(1+0)+sqrt(1+0))) = 2/2=1`

As a check, here's the graph of the function `f(x) = (sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x))`

graph{(sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x)) [-8.794, 8.98, -1.626, 7.264]}

Answer 2

`1`

Explanation:

`(sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x))`

Plugging in zero gives:

`(sqrt(1+(0)^2)-sqrt(1+0))/(sqrt(1+(0)^3)-sqrt(1+0))=0/0`

Using L'Hospital's Rule:

Differentiate numerator and denominator.

Numerator:

`d/dx((1+x^2)^(1/2)-(1+x)^(1/2))=1/2(1+x^2)^(-1/2)*2x-1/2(1+x)^(-1/2)*1=color(blue)(x/(1+x^2)^(1/2)-1/(2(1+x)^(1/2))`

Denominator:

`d/dx((1+x^3)^(1/2)-(1+x)^(1/2))=1/2(1+x^3)^(-1/2)*3x^2-1/2(1+x)^(-1/2)*1=color(blue)((3x^2)/(2(1+x^3)^(1/2))-1/(2(1+x)^(1/2))`

`(x/(1+x^2)^(1/2)-1/(2(1+x)^(1/2)))/((3x^2)/(2(1+x^3)^(1/2))-1/(2(1+x)^(1/2))`

Plugging in zero:

`(0/(1+(0)^2)^(1/2)-1/(2(1+0)^(1/2)))/((3(0)^2)/(2(1+(0)^3)^(1/2))-1/(2(1+0)^(1/2)))=(-1/2)/(-1/2)=1`

`:.`

`lim_(x->0)((sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x)))=1`