𝑛→+∞ (︁4^√𝑛^4 + 2𝑛^3 - 4^√𝑛^4 + 𝑛^3)︁

Let's analyze the given expression as \( n \to +\infty \):

\[
\lim_{n \to +\infty} \left( 4^{\sqrt{n^4}} + 2n^3 - 4^{\sqrt{n^4}} + n^3 \right)
\]

First, simplify the expression \( 4^{\sqrt{n^4}} \).

### Step 1: Simplify \( 4^{\sqrt{n^4}} \)

Recall that:

\[
\sqrt{n^4} = n^2
\]

Thus,

\[
4^{\sqrt{n^4}} = 4^{n^2}
\]

### Step 2: Substitute into the expression

Now substitute \( 4^{n^2} \) into the original expression:

\[
\lim_{n \to +\infty} \left( 4^{n^2} + 2n^3 - 4^{n^2} + n^3 \right)
\]

This simplifies to:

\[
\lim_{n \to +\infty} \left( 2n^3 + n^3 \right)
\]

### Step 3: Combine like terms

Now, add the terms \( 2n^3 + n^3 \):

\[
\lim_{n \to +\infty} 3n^3
\]

### Step 4: Evaluate the limit

As \( n \to +\infty \), \( 3n^3 \to +\infty \). Thus, the final result is:

\[
\boxed{+\infty}
\]

Would you like more details or have any questions? 

Here are some related questions you might find helpful:
1. How do limits behave for different powers of \( n \)?
2. Why does \( 4^{n^2} \) cancel out in the expression?
3. What are common forms of infinity in limits?
4. How does \( n^3 \) compare with higher-order terms like \( 4^{n^2} \)?
5. How would the result change if the powers of \( n \) were different?

**Tip:** When dealing with limits at infinity, focus on the terms with the largest growth rate, as they often dominate the result.

Discover how to evaluate the limit of a complex expression involving exponential and polynomial terms as n approaches infinity. Learn the steps to simplify and understand growth rates of functions in this calculus problem, suitable for advanced students.

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