# Thread: Solving for a square root limit

1. ## Solving for a square root limit

When I try to solve limits I find I always get the answer wrong when dealing with square roots.

Here's an example:

lim x --> - infinity (sqrt(x^2 + x + 1) + x

what I do:

lim x --> - infinity (sqrt(x^2 + x + 1) + x (((sqrt(x^2+x+1) - x)/((sqrt(x^2+x+1) - x))

lim x --> -infinity (x^2 + x + 1 - x^2)/(sqrt(x^2 + x + 1) - x)

lim x --> -infinity (x + 1)/(sqrt(x^2 + x + 1) - x)

lim x --> -infinity (x + 1)/(sqrt(x^2( 1 + 1/x + 1x^2) - x)

lim x --> -infinity (x/x(1 + 1/x)/(sqrt((x^2)/(x^2)( 1 + 1/x + 1x^2) - x/x)

so I get 1/sqrt(1-1) = Does not exist

What am I doing wrong?

Any help would be greatly appreciated!

2. -delete-

3. Originally Posted by s3a
When I try to solve limits I find I always get the answer wrong when dealing with square roots.

Here's an example:

lim x --> - infinity (sqrt(x^2 + x + 1) + x

what I do:

lim x --> - infinity (sqrt(x^2 + x + 1) + x (((sqrt(x^2+x+1) - x)/((sqrt(x^2+x+1) - x))

lim x --> -infinity (x^2 + x + 1 - x^2)/(sqrt(x^2 + x + 1) - x)

lim x --> -infinity (x + 1)/(sqrt(x^2 + x + 1) - x)

lim x --> -infinity (x + 1)/(sqrt(x^2( 1 + 1/x + 1x^2) - x)

lim x --> -infinity (x/x(1 + 1/x)/(sqrt((x^2)/(x^2)( 1 + 1/x + 1x^2) - x/x)
Here's your error. $\sqrt{x^2}= |x|$, not x. And since you are taking x going to negative infinity, $\sqrt{x^2}= -x$. You should have $1/(\sqrt{1}+ 1)$

so I get 1/sqrt(1-1) = Does not exist

What am I doing wrong?

Any help would be greatly appreciated!

4. Originally Posted by s3a
When I try to solve limits I find I always get the answer wrong when dealing with square roots.

Here's an example:

lim x --> - infinity (sqrt(x^2 + x + 1) + x

what I do:

lim x --> - infinity (sqrt(x^2 + x + 1) + x (((sqrt(x^2+x+1) - x)/((sqrt(x^2+x+1) - x))

lim x --> -infinity (x^2 + x + 1 - x^2)/(sqrt(x^2 + x + 1) - x)

lim x --> -infinity (x + 1)/(sqrt(x^2 + x + 1) - x)

lim x --> -infinity (x + 1)/(sqrt(x^2( 1 + 1/x + 1x^2) - x)

lim x --> -infinity (x/x(1 + 1/x)/(sqrt((x^2)/(x^2)( 1 + 1/x + 1x^2) - x/x)

so I get 1/sqrt(1-1) = Does not exist

What am I doing wrong?

Any help would be greatly appreciated!
$\mathop {\lim }\limits_{x \to - \infty } \left( {\sqrt {{x^2} + x + 1} + x} \right) = \mathop {\lim }\limits_{x \to - \infty } \frac{{\left( {\sqrt {{x^2} + x + 1} + x} \right)\left( {\sqrt {{x^2} + x + 1} - x} \right)}}
{{\sqrt {{x^2} + x + 1} - x}} =$

$= \mathop {\lim }\limits_{x \to - \infty } \frac{{x + 1}}{{\sqrt {{x^2} + x + 1} - x}} = \mathop {\lim }\limits_{x \to - \infty } \frac{{x + 1}}{{\left| x \right|\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} - x}} =$

$= \mathop {\lim }\limits_{x \to - \infty } \frac{{x\left( {1 + \frac{1}
{x}} \right)}}{{ - x\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} - x}} = \mathop {\lim }\limits_{x \to - \infty } \frac{{1 + \frac{1}{x}}}{{ - \sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} - 1}} =$

$= \frac{{1 - 0}}{{ - \sqrt {1 - 0 - 0} - 1}} = \frac{1}{{ - 1 - 1}} = - \frac{1}{2}.$

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### Lim x->1 ((sqrt x - 1)/(x - 1))

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