Certainly! I will first generate a similar problem and provide its solution. Then, I will give you a set of five similar problems (with no solutions) for practice. Let’s start with one problem and a step-by-step solution to guide you.

New Problem 1

Evaluate the limit using the series expansion:
$\lim_{x \to 0} \frac{\cos(3x) - 1 + \frac{9}{2}x^2}{x^4}.$

Solution

To solve this limit, we use the Taylor series expansion of $\cos(x)$ near $x = 0$:

$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \mathcal{O}(x^6).$

1. Substitute $3x$ into the series for $\cos(3x)$: $\cos(3x) = 1 - \frac{(3x)^2}{2} + \frac{(3x)^4}{24} + \mathcal{O}(x^6).$ Simplify the terms: $\cos(3x) = 1 - \frac{9x^2}{2} + \frac{81x^4}{24} + \mathcal{O}(x^6).$

2. Subtract 1 and add $\frac{9}{2}x^2$: $\cos(3x) - 1 + \frac{9}{2}x^2 = \left(1 - \frac{9x^2}{2} + \frac{81x^4}{24}\right) - 1 + \frac{9}{2}x^2 + \mathcal{O}(x^6).$ Combine like terms: $\cos(3x) - 1 + \frac{9}{2}x^2 = \frac{81x^4}{24} + \mathcal{O}(x^6).$

3. Divide by $x^4$: $\frac{\cos(3x) - 1 + \frac{9}{2}x^2}{x^4} = \frac{\frac{81x^4}{24} + \mathcal{O}(x^6)}{x^4}.$ Simplify: $\frac{\cos(3x) - 1 + \frac{9}{2}x^2}{x^4} = \frac{81}{24} + \mathcal{O}(x^2).$

4. Take the limit as $x \to 0$: As $x \to 0$, the higher-order terms vanish, so: $\lim_{x \to 0} \frac{\cos(3x) - 1 + \frac{9}{2}x^2}{x^4} = \frac{81}{24} = \frac{27}{8}.$

Final Answer:

$\frac{27}{8}.$

Practice Problems

1. Evaluate the limit using series:
   $\lim_{x \to 0} \frac{\sin(3x) - 3x + \frac{9}{2}x^3}{x^5}.$

2. Evaluate the limit:
   $\lim_{x \to 0} \frac{\cos(2x) - 1 + 2x^2}{x^4}.$

3. Evaluate the limit:
   $\lim_{x \to 0} \frac{\sin(5x) - 5x + \frac{125}{6}x^3}{x^5}.$

4. Use series expansion to find:
   $\lim_{x \to 0} \frac{\cos(4x) - 1 + 8x^2}{x^4}.$

5. Evaluate the limit:
   $\lim_{x \to 0} \frac{\sin(2x) - 2x + \frac{8}{3}x^3}{x^5}.$

Tip:

Always expand trigonometric functions like $\sin(x)$ and $\cos(x)$ into their Taylor series up to the required order. Then, carefully combine terms and divide to simplify the limit.

Would you like solutions to any of these problems? Let me know! 😊