Question

lim ((1-2x)/(√1+x^2)-1)
x→∞

Find the value.

Answer 1

you have

(1-2x)/(√(1+x^2)-1)
multiply top and bottom to get

(1-2x)(√(1+x^2)+1) / (√(1+x^2)-1)(√(1+x^2)+1)

= (1-2x)(√(1+x^2)+1)/x^2
= (1-2x+√(1+x^2)-2x√(1+x^2))/x^2
= 1/x^2 - 2/x + √(1/x^4+1/x^2) - 2√(1+1/x^2)

as x→∞, that is 0-0+0-2 = -2