Provide me Expansion based problem on limit questions

Certainly! Below is an expansion-based problem on limits:

### Problem:
Evaluate the limit using the binomial expansion:
\[
\lim_{{x \to 0}} \frac{{(1 + 2x)^{5} - (1 + 3x)^{4}}}{x}
\]

### Solution Outline:
1. **Expand** both \((1 + 2x)^5\) and \((1 + 3x)^4\) using the binomial theorem.
2. **Subtract** the expanded forms.
3. **Simplify** the expression by canceling terms.
4. **Divide** the resulting expression by \(x\).
5. **Evaluate** the limit as \(x \to 0\).

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### Related Questions:
1. Evaluate \(\lim_{{x \to 0}} \frac{{(1 + x)^{6} - (1 - x)^{6}}}{x}\).
2. Find \(\lim_{{x \to 0}} \frac{{(1 + 4x)^{3} - (1 + 2x)^{4}}}{x}\) using expansion.
3. Evaluate \(\lim_{{x \to 0}} \frac{{(1 + 2x)^{7} - (1 + 5x)^{2}}}{x}\).
4. Expand and find \(\lim_{{x \to 0}} \frac{{(1 + x)^{8} - (1 + 2x)^{4}}}{x}\).
5. Determine \(\lim_{{x \to 0}} \frac{{(1 + 3x)^{5} - (1 + x)^{5}}}{x}\) using binomial expansion.

### Tip:
When solving limit problems using binomial expansion, it's important to expand terms up to the power of \(x\) that ensures terms higher than \(x\) cancel out, simplifying the limit calculation.

Learn how to evaluate the limit using binomial expansion: (1 + 2x)^5 - (1 + 3x)^4 / x. This problem involves applying the binomial theorem to expand and simplify expressions, suitable for advanced high school and college-level students.

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