To find the limit of the given expression as x approaches 0, we can use direct substitution:

lim x->0 (x^2 + x) / √(x^4 + 2x^2)





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.

To find the limit of the given expression as x approaches 0, we can use direct substitution:
lim x->0 (x^2 + x) / √(x^4 + 2x^2)

Explore how to evaluate the limit of (x^2 + x) / √(x^4 + 2x^2) as x approaches 0. This detailed explanation breaks down the steps using limit laws, indeterminate forms, and sign function properties. Suitable for college-level students studying calculus.

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