Limit of trigometric function with x-sqrt/x-sqrt

[mesasi]

Homework Statement

Find the value of lim x→∞

[x-sqrt(x^2+5x+2)] / [x-sqrt(x^2+(x/2)+1)]

Answer is 10.

Homework Equations

The Attempt at a Solution

I tried to multiply by the conjugate and got

[5x+2] / [x-sqrt(x^2+5x+2)] * [x+sqrt(x^2+(x/2)+1)]

but then I'm still stuck because I still get ∞ as the answer.

I also tried to divide both the top and bottom by x. Then I get [1-sqrt(1+(5/x)-(2/x^2))]/[1-sqrt(1+(x^3/2)+(1/x^2)) = 0/0 which is incorrect

 

[SammyS]

Homework Statement

Find the value of lim x→∞

[x-sqrt(x^2+5x+2)] / [x-sqrt(x^2+(x/2)+1)]

Answer is 10.

Homework Equations

The Attempt at a Solution

I tried to multiply by the conjugate and got

[5x+2] / [x-sqrt(x^2+5x+2)] * [x+sqrt(x^2+(x/2)+1)]

but then I'm still stuck because I still get ∞ as the answer.

I also tried to divide both the top and bottom by x. Then I get [1-sqrt(1+(5/x)-(2/x^2))]/[1-sqrt(1+(x^3/2)+(1/x^2)) = 0/0 which is incorrect

Hello mesasi. Welcome to PF !

Also multiply the numerator & denominator by the conjugate of [x-sqrt(x^2+(x/2)+1)]

 

[mesasi]

After I do and divide by x on both sides I get





\frac{(-5x-2)(1+\sqrt{1+(1/2x)+(1/x^2)}}{((-x/2)-1)(1+\sqrt{1+(5/x)+(2/x^2)}}


then I get \frac{-5x}{(-x/2)}+2

which simplifies to 12 which is still not 10? What am I doing wrong?

 

[SammyS]

After I do and divide by x on both sides I get

\frac{(-5x-2)(1+\sqrt{1+(1/2x)+(1/x^2)}}{((-x/2)-1)(1+\sqrt{1+(5/x)+(2/x^2)}}

then I get \frac{-5x}{(-x/2)}+2

which simplifies to 12 which is still not 10? What am I doing wrong?

\displaystyle \frac{-5x-2}{1+(1/2x}\ne\frac{-5x}{(-x/2)}+2

 

[mesasi]

(5x+2)/((x/2)+1) * (2/2) = 10

Thank you!