To find the limit of the expression $\lim_{{x \to 0}} \frac{{x^2 + x}}{{\sqrt{x^4 + 2x^2}}}$, let's break it down step by step.

Step 1: Direct Substitution

First, substitute $x = 0$ directly into the expression:

$\frac{{0^2 + 0}}{{\sqrt{0^4 + 2(0^2)}}} = \frac{0}{\sqrt{0 + 0}} = \frac{0}{0}$

This results in an indeterminate form $\frac{0}{0}$, so direct substitution doesn't give us the limit.

Step 2: Simplification of the Expression

To resolve the indeterminate form, let's simplify the expression by factoring and analyzing its components.

1. The numerator $x^2 + x$ can be factored as: $x(x + 1)$

2. The denominator $\sqrt{x^4 + 2x^2}$ can be simplified by factoring out $x^2$ from under the square root: $\sqrt{x^4 + 2x^2} = \sqrt{x^2(x^2 + 2)} = |x|\sqrt{x^2 + 2}$

So, the expression now becomes: $\frac{x(x + 1)}{|x|\sqrt{x^2 + 2}}$

Step 3: Further Simplification

We can simplify the fraction by canceling out the $x$ term (but remember to treat the absolute value properly):

$\frac{x(x + 1)}{|x|\sqrt{x^2 + 2}} = \frac{x + 1}{\text{sgn}(x)\sqrt{x^2 + 2}}$

Where $\text{sgn}(x)$ denotes the sign function, which equals 1 when $x > 0$ and -1 when $x < 0$.

Step 4: Taking the Limit

Now, let's take the limit as $x$ approaches 0:

$\lim_{{x \to 0}} \frac{x + 1}{\text{sgn}(x)\sqrt{x^2 + 2}}$

As $x \to 0$, $\sqrt{x^2 + 2}$ approaches $\sqrt{2}$, and the expression simplifies to:

For $x > 0$: $\lim_{{x \to 0^+}} \frac{0 + 1}{1\cdot\sqrt{2}} = \frac{1}{\sqrt{2}}$

For $x < 0$: $\lim_{{x \to 0^-}} \frac{0 + 1}{-1\cdot\sqrt{2}} = \frac{1}{-\sqrt{2}} = -\frac{1}{\sqrt{2}}$

Step 5: Conclusion

The left-hand limit and the right-hand limit are not equal. Therefore, the overall limit does not exist as $x$ approaches 0.

Would you like any further details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the significance of the sign function in limits involving absolute values?
2. How do you apply L'Hôpital's Rule to solve indeterminate forms like $\frac{0}{0}$?
3. How can you determine the continuity of a function at a point where the limit does not exist?
4. What are the conditions under which a limit might not exist?
5. How do you evaluate limits that involve piecewise functions?

Tip: When dealing with indeterminate forms, always explore simplifying the expression or applying limit laws before jumping to advanced techniques like L'Hôpital's Rule.