What am I doing wrong when evaluating this limit?

[tahayassen]

\underset { x\rightarrow -\infty }{ lim } (\sqrt { { x }^{ 2 }+x+1 } +x)\\ =\underset { x\rightarrow -\infty }{ lim } (|x|\sqrt { 1+{ x }^{ -1 }+{ x }^{ -2 } } +x)\\ Since\quad x\rightarrow -\infty \\ =\underset { x\rightarrow -\infty }{ lim } (-x\sqrt { 1+{ x }^{ -1 }+{ x }^{ -2 } } +x)\\ =\underset { x\rightarrow -\infty }{ lim } x(-\sqrt { 1+{ x }^{ -1 }+{ x }^{ -2 } } +1)\\ =\underset { x\rightarrow -\infty }{ lim } x\quad *\underset { x\rightarrow -\infty }{ lim } (-\sqrt { 1+{ x }^{ -1 }+{ x }^{ -2 } } +1)\\ =\quad -\infty *0\\ =\quad 0

Before you say that you can't multiply infinity by 0, why not? If we thinking infinity as a very large number, it doesn't matter how large it is, if it gets multiplied by 0, it will equal 0, right?

 

[arildno]

Your limiting operation is perfectly OKAY, but your conclusion is still wrong.
Basically, because "infinity" is not a number, and therefore cannot be multiplied reliable with any number.
The critical issue is what limit the FINITE products tend to.
And THAT depends on the RATES by which one of the factors tend to zero, the other to negative infinity.
The result should be -1/2

 

[LCKurtz]

Before you say that you can't multiply infinity by 0, why not? If we thinking infinity as a very large number, it doesn't matter how large it is, if it gets multiplied by 0, it will equal 0, right?

No. That's wrong. $0\cdot\infty$ is an indeterminate form just like $\frac 0 0$ is. Look at$$
\lim_{x\rightarrow 0} x\cdot \frac 1 x$$ That is a $0\cdot\infty$ form and its limit is 1. You can make it be anything by putting a C in there.

 

[tahayassen]

Ah, makes sense now. Thank you.

 

[arildno]

In effect, your ugly factor will look more and more like -1/2*1/x, as x tends to negative infinity.
Thus, multiplying THIS with "x" gives you the correct limiting expression.

 

[tahayassen]

In effect, your ugly factor will look more and more like -1/2*1/x, as x tends to negative infinity.
Thus, multiplying THIS with "x" gives you the correct limiting expression.

Where did you get -1/2*1/x?

 

[arildno]

Multiply the ugly factor with its own conjugate.
And..simplify!

 

[Mark44]

Multiply the ugly factor with its own conjugate.
And..simplify!

I think you mean, multiply the "ugly factor" by 1 in the form of the conjugate over itself.

 

[arildno]

I think you mean, multiply the "ugly factor" by 1 in the form of the conjugate over itself.

Reluctantly, and with smoke pouring out of my nostrils, I have to agree..

 

[Mark44]

I knew what you meant...

 

[arildno]

I knew what you meant...

That doesn't lead to smoke reduction, since it is the effect of self loathing ..