#"Your "color(red)"Question:"#
#color(red)(√(x^2 +1) -1 / √(x^2+16) - 4#
#"Answer for the "color(blue)" Question:"#
#color(blue)(lim_(xto0)(sqrt(x^2+1)-1)/(sqrt(x^2+16)-4)#
#lim_(xto0)
[(sqrt(x^2+1)-1)/(sqrt(x^2+16)-4)xx(sqrt(x^2+1)+1)/(sqrt(x^2+1)+
1)xx(sqrt(x^2+16)+4)/(sqrt(x^2+16)+4)]#
#=lim_(xto0)
[((x^2+1)-1)/(sqrt(x^2+1)+1)xx(sqrt(x^2+16)+4)/((x^2+16)-16)]#
#=lim_(xto0)
[((cancelx^2)/(sqrt(x^2+1)+1)xx(sqrt(x^2+16)+4)/(cancelx^2)]#
#=1/(sqrt(0+1)+1)xx(sqrt(0+16)+4)/1#
#=1/2xx8#
#=4#