Exercise 2 Using the following Taylor Polynomial expansions...

Exercise 2 Using the following Taylor Polynomial expansions at x = 0 $\log(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^n}{n} + o(x^n)$ $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} + o(x^n)$ $\sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 + o(x^3)$, compute the following limit $\lim_{x\to 0} \frac{x^3e^x - \ln(1+x^3)}{(\sqrt{1+x^2}-1)^2}$

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Exercise 2: Using the following Taylor Polynomial expansions at x = 0, compute the following limit: 1. log(1 + x): - Taylor Polynomial expansion: Σ((-1)^h * x^(h+1))/(h+1), h=0 to n - Limit: lim(x→0) log(1 + x) 2. e^(√(1 + x)): - Taylor Polynomial expansion: Σ(x^h)/(h!), h=0 to n - Limit: lim(x→0) e^(√(1 + x)) 3. ln(1 + x^3): - Taylor Polynomial expansion: Σ((-1)^h * x^(3h+1))/(3h+1), h=0 to n - Limit: lim(x→0) ln(1 + x^3) 4. (√(1 + x^2) - 1)^2: - Taylor Polynomial expansion: (x^3) + 0(x^3) - Limit: lim(x→0) (√(1 + x^2) - 1)^2 Note: Please correct any spelling, typographical, grammatical, OCR (optical character recognition), and mathematical errors, including any errors related to the square root symbol. Ensure that the entire text is properly formatted and presented in a clear, coherent manner. Only correct errors, do not answer it.

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