Question #7c54f

2 Answers
Dec 29, 2017

see below

Explanation:

Lim_(xrarr0)sqrt(x²+2x+2)-x=sqrt2

or

Lim_(xrarroo)sqrt(x²+2x+2)-x*(sqrt(x²+2x+2)+x)/(sqrt(x²+2x+2)+x)

Lim_(xrarroo)(x²+2x+2-x^2)/(sqrt(x²+2x+2)+x)

Lim_(xrarroo)(2x+2)/(sqrt(x²+2x+2)+x)

Lim_(xrarroo)(cancelx(2+2/x))/(cancelx(sqrt(1+2/x+2/x^2)+1)

Lim_(xrarroo)(2+2/x)/(sqrt(1+2/x+2/x^2)+1)=2/(1+1)=1

You have a mistake but a correct solution :D

Dec 29, 2017

You have the basic ideas but there are some errors in your details (and these errors canceld out so you got the correct answer.)

Explanation:

((sqrt(x²+2x+2)-x))/1 * ((sqrt(x²+2x+2)+x))/((sqrt(x²+2x+2)+x)) = ((x^2+2x+2)-x^2)/(sqrt(x²+2x+2)+x)

= (2x+2)/(sqrt(x^2(1+2/x+2/x^2))+x) " " for x != 0

= (2x+2)/(sqrt(x^2)sqrt(1+2/x+2/x^2)+x)

Use sqrt(x^2) = absx and x > 0 (so absx=x) to get

= (2x+2)/(xsqrt(1+2/x+2/x^2)+x)

= (x(2+2/x))/(x(sqrt(1+2/x+2/x^2)+1))

= (2+2/x)/(sqrt(1+2/x+2/x^2)+1)

Now find the limit (I assume we are finding the limit at infinity)

lim_(xrarroo)(sqrt(x²+2x+2)-x) = lim_(xrarroo)(2+2/x)/(sqrt(1+2/x+2/x^2)+1)

= (2+0)/(sqrt(1+0+0)+1) = 2/2=1

Note that
Your 2x^2 in the numerator is incorrect

and

lim_(xrarr00)(2x+2+(2/x))/(sqrt(1+(2/x)+(2/x²))+1) = (oo+2+0)/(sqrt1+1) = oo " "

The limit of your expression is not " " 2/2=1