Esercitazione guidata Analisi Matematica 1 - sesta parte

1)

\(\lim_{{x \to 0^+}} \frac{{x \sin x - \cos x + \alpha}}{{\text{arctg}(x^2)}}\)

\(= \lim_{{x \to 0^+}} \frac{{x(x + o(x)) - \left(1 - \frac{1}{2} x^2 + o(x^2)\right) + \alpha}}{{x^2 + o(x^2)}}\)

\(= \lim_{{x \to 0^+}} \frac{{x^2 + o(x^2) - 1 + \frac{1}{2} x^2 + o(x^2) + \alpha}}{{x^2 + o(x^2)}}\)

\(= \lim_{{x \to 0^+}} \frac{{\alpha - 1 + \frac{3}{2} x^2 + o(x^2)}}{{x^2 + o(x^2)}}\)

\(= \left\{ \begin{array}{ll} \text{se } \alpha - 1 = 0 & \frac{3}{2} \\ \text{se } \alpha - 1 > 0 & +\infty \\ \text{se } \alpha - 1

2)

\(\lim_{{x \to 0^+}} \frac{{\sin x - \cos x}}{{x + \log(1 + x^\alpha)}},\; \alpha > 0\)

\(= \lim_{{x \to 0^+}} \frac{{x + o(x) - x \left(1 - \frac{1}{2} x^2 + o(x^2)\right)}}{{x + x^\alpha + o(x^\alpha)}}\)

\(= \lim_{{x \to 0^+}} \frac{{x + o(x) - x + \frac{1}{2} x^3 + o(x^3)}}{{x + x^\alpha + o(x^\alpha)}}\)

\(= \lim_{{x \to 0^+}} \frac{{\cancel{x} + x^\alpha + o(x^\alpha)}}{{\cancel{x} + \frac{1}{2} x^3 + o(x^3)}}\)

\(x^\alpha + x^\alpha + o(x^\alpha) = \left\{ \begin{array}{ll} \text{se } \alpha = 1 & 2x + o(x) \to 0 \\ \text{se } \alpha > 1 & x\left(1 + x^{\alpha - 1} + o(x^{\alpha - 1})\right) \\ \text{se } \alpha

1)

limx→0+ xsinx - cosx + α/arctg(x2)= limx→0+ x(x + o(x)) - (1 - 1/2x2 + o(x2)) + α/x2 + o(x2)= limx→0+ x2 + o(x2) - 1 + 1/2x2 + o(x3) + α/x2 + o(x2)= limx→0+ α-1 + 3/2x2 + o(x2)/x2 + o(x2)= se α-1 = 0     3/2se α-1 > 0     +∞se α-1

2)

limx→0+ sinx - cosx/x + log(1 + xα) ,  α > 0= limx→0+ x + o(x) - x(1/2 - 1/2x2 + o(x2))/x + xα + o(xα)= limx→0+ x + o(x) - x + 1/2x3 + o(x3)/x + xα + o(xα)= limx→0+ 1/2x3 + o(x3)/x + xα + o(xα)x + xα + o(xα) = se α=1     2x + o(x) → 0se α>1     x(1 + xα-1 + o(xα-1))se α