How to solve lim_(x->0)((1/(x(sqrt(x+1))))-1/x)?

1 Answer
Dec 28, 2017

Please see below.

Explanation:

Get a single quotient, then rationalize the numerator.

1/(xsqrt(x+1))-1/x = (1-sqrt(x+1))/(xsqrt(x+1))

= ((1-sqrt(x+1)))/(xsqrt(x+1)) * ((1+sqrt(x+1)))/ ((1+sqrt(x+1)))

= (1-x-1)/(xsqrt(x+1)(1+sqrt(x+1)))

= (-1)/(sqrt(x+1)(1+sqrt(x+1)))

lim_(xrarr0)(1/(xsqrt(x+1))-1/x) = lim_(xrarr0)(-1)/(sqrt(x+1)(1+sqrt(x+1)))

= (-1)/(sqrt1(1+sqrt1)) = -1/2

We don't need it but here is the graph:

graph{1/(xsqrt(x+1))-1/x [-3.33, 6.537, -3.61, 1.322]}