How do you find the limit of #sqrt(x+e^(4x))/(e^(2x)+x)# as #x->oo#?

1 Answer
Nov 29, 2016

#lim_(x->oo) sqrt(x+e^(4x))/(e^(2x)+x) =1#

Explanation:

Bring the denominator under the square root:

#f(x) = sqrt(x+e^(4x))/(e^(2x)+x) = sqrt(((x+e^(4x))/(e^(2x)+x)^2)#

Expand:

#f(x) = sqrt(((x+e^(4x))/(e^(4x)+2xe^(2x)+x^2))#

Divide by #e^(4x)# above and below:

#f(x) = sqrt(((xe^(-4x)+1)/(1+2xe^(-2x)+x^2e^(-4x)))#

As for every #n# and #alpha>0#: #lim_(x->oo) x^n e^(-alpha x) = 0#

#lim_(x->oo) f(x) = lim_(y->1) sqrt(y) = 1#