Uppgift 1

a) Förläng med konjugatuttrycket:

$$\sqrt{n^2+an+b}-n=\frac{an+b}{\sqrt{n^2+an+b}+n}=\frac{a+b/n}{\sqrt{1+a/n+b/n^2}+1}.$$

Alltså

$$\lim_{n\to\infty}(\sqrt{n^2+an+b}-n)=\frac a2.$$

b) Standardutvecklingarna

$$e^t=1+t+O(t^2),\quad \cos t=1-\frac{t^2}{2}+O(t^3),\quad \ln(1+t)=t+O(t^2)$$

ger

$$\frac{e^{x^2}-\cos(2x)}{x\ln(1+4x)}=\frac{(1+x^2+O(x^4))-(1-2x^2+O(x^3))}{x(4x+O(x^2))}$$

$$=\frac{3+O(x)}{4+O(x)}\longrightarrow\frac34.$$