Homework Sourselim as xInfinity 4x2x5 2xSolution limxinfinity sqrt4 x2x2 x
lim as x-->Infinity
((4x^2+x)^.5) -2x
((4x^2+x)^.5) -2x
Solution
lim_(x->infinity) (sqrt(4 x^2+x)-2 x) Rationalize sqrt(4 x^2+x)-2 x = (sqrt(4 x^2+x)-2 x) (2 x+sqrt(x+4 x^2))/(2 x+sqrt(x+4 x^2)) = x/(sqrt(4 x^2+x)+2 x): = lim_(x->infinity) x/(sqrt(4 x^2+x)+2 x) = lim_(x->infinity) 1/(sqrt(4 x^2+x)/x+2) = 1/(lim_(x->infinity) (sqrt(4 x^2+x)/x+2)) = 1/(lim_(x->infinity) sqrt(4 x^2+x)/x+2) = 1/(lim_(x->infinity) sqrt((4 x^2+x)/x^2)+2) = 1/(sqrt(lim_(x->infinity) (4 x^2+x)/x^2)+2) = 1/(sqrt(lim_(x->infinity) (4+1/(2 x)))+2) = 1/(sqrt(1/2 (lim_(x->infinity) 1/x)+4)+2) = 1/(sqrt(1/(2 (lim_(x->infinity) x))+4)+2) The limit of x as x approaches infinity is infinity: = 1/4
