\lim _{x\to 0}\left(\frac{x^3+4x^4-2x^2}{x^2}\right)

We are asked to compute the following limit:

\[
\lim _{x\to 0}\left(\frac{x^3+4x^4-2x^2}{x^2}\right)
\]

### Step-by-step solution:

1. **Simplify the expression**: First, we can simplify the given expression by factoring \(x^2\) out from both the numerator and the denominator:
   \[
   \frac{x^3+4x^4-2x^2}{x^2} = \frac{x^2(x + 4x^3 - 2)}{x^2}
   \]
   Cancelling \(x^2\) from the numerator and the denominator, we are left with:
   \[
   x + 4x^3 - 2
   \]

2. **

Explore how to calculate the limit \lim _{x\to 0}\left(\frac{x^3+4x^4-2x^2}{x^2}\right). This problem covers essential calculus concepts such as limits and algebraic simplification, suitable for students in Grades 11-12.

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