Question

Apparent discrepancy - Resolve the apparent discrepancy between:

a) Integral dx/x(x-1)(x²)
b) Derivative of x³
c) Limit of (x-1)/x as x approaches infinity
d) Taylor series expansion of ln(x)

Answer 1

The derivative of x³ is 3x². The limit of (x-1)/x as x approaches infinity is 1. The Taylor series expansion of ln(x) is given by a series involving (-1)^(n+1) and (x-1)^n.

Step-by-step explanation:

To resolve the apparent discrepancy, we need to analyze each part separately:

1. For the integral ∫ dx/x(x-1)(x²), we can factor the denominator to rewrite it as ∫ dx/(x²)(x-1). We then use partial fraction decomposition to split the integrand into partial fractions, and integrate each term separately.
2. The derivative of x³ is simply 3x² by applying the power rule of differentiation.
3. The limit of (x-1)/x as x approaches infinity can be evaluated by dividing each term by the highest power of x in the denominator. In this case, the limit is 1.
4. The Taylor series expansion of ln(x) is given by Σ(-1)^(n+1)(x-1)^n/n, where the sum is taken from n = 1 to infinity.