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

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Answer 1 · 2017-12-28T04:26:31

Please see below.

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]}