Lim x-> 0+ [sqrt(x^2 + 4x + 5) - sqrt(5)] / x?

Question

Find the limit please! And explain how you got it.

Answer 1

on multiplying with sqrt[x^2+4x+5]+sqrt[5] at Nr and Dr

lim x-->o+ [x^2+4x]/x[sqrt[x^2+4x+5]+sqrt[5]]

=limx-->o+ [x+4]/[sqrt[x^2+4x+5]+sqrt[5]]

=4/2sqrt(5) = 2/sqrt(5)

Answer 2

this is in the 0/0 form so we can use L'hopitals

lim x-> 0+ [√(x² + 4x + 5) - √(5)] / x

= lim x-> 0+ (2x + 4)/2√(x² + 4x + 5)

= 4/2√(5)

= 2/√(5)

= 2√(5)/5 (if you want a rational denominator)

Answer 3

{x+sqrt x^2+4x+5-sqrt 5}/x

=1+{sqrt x^2+ 4x+5- sqrt5}/x

= 1+{sqrt x^2+4x+5 - sqrt 5*sqrt x^2+4x+5 + sqrt5}/x* sqrt 5*sqrt x^2+4x+5 + sqrt5}

= 1+x+4/ sqrt x^2+4x+5 +sqrt 5

= 1+2/sqrt 5