Question

How do you find the limit of `(x^2 -sqrt x)/(sqrt x -1)` as x approaches 1?

Answer 1

`lim_(xrarr1)(x^2-sqrtx)/(sqrtx-1)=3`

Explanation:

Note that this is in the indeterminate form `0/0`, which means we can apply L'Hopital's rule. This means the limit will be equivalent if we take the derivative of the numerator over the derivative of the denominator.

`lim_(xrarr1)(x^2-sqrtx)/(sqrtx-1)`

Apply L'Hopital's:

`=lim_(xrarr1)(d/dx(x^2-sqrtx))/(d/dx(sqrtx-1))=lim_(xrarr1)(2x-1/(2sqrtx))/(1/(2sqrtx))`

Simplifying:

`=lim_(xrarr1)(2x-1/(2sqrtx))(2sqrtx)=lim_(xrarr1)(4x^(3/2)-1)`

Now evaluating the limit:

`=4(1)-1=3`