Multiply and divide the function by:
#(sqrt(x^2+1)+1)/(sqrt(x^2+16)+4)#
and use the algebraic identity:
#(a-b)(a+b) = a^2-b^2#
to have:
#lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) = lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) xx (sqrt(x^2+1)+1)/(sqrt(x^2+16)+4) xx (sqrt(x^2+16)+4)/(sqrt(x^2+1)+1)#
#lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) = lim_(x->0) (x^2+1-1)/(x^2+16-16) xx (sqrt(x^2+16)+4)/(sqrt(x^2+1)+1)#
#lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) = lim_(x->0) x^2/x^2 xx (sqrt(x^2+16)+4)/(sqrt(x^2+1)+1)#
#lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) = lim_(x->0) (sqrt(x^2+16)+4)/(sqrt(x^2+1)+1)#
#lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) = (sqrt(16)+4)/(sqrt(1)+1)#
#lim_(x->0) (sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) = 4#
graph{(sqrt(x^2+1)-1)/(sqrt(x^2+16)-4) [-10, 10, -5, 5]}