Math  /  Calculus

QuestionExercise 1. Determine limited development in the neighborhood of x0=0x_{0}=0 to the indicated order nn for the following functions
1. 1+x+x21+x+x3(n=3)I\frac{1+x+x^{2}}{1+x+x^{3}} \quad(n=3){ }^{I}
3. arctan(2x)1+sinx(n=3)\frac{\arctan (2 x)}{-1+\sin x} \quad(n=3),
2. 1+x2x2(n=2)\sqrt{\frac{1+x}{2-x^{2}}} \quad(n=2)
4. sh2xln(cosx)(n=2)\frac{\operatorname{sh}^{2} x}{\ln (\cos x)} \quad(n=2),
5. e2+cosx(n=2)e^{\sqrt{2+\cos x}} \quad(n=2)

Exercise 2.
1. Determine the limited development in the neighborhood of x0x_{0} to the indicated order nn of the following functions : (a) f(x)=cosx,x0=π4,n=3f(x)=\cos x, x_{0}=\frac{\pi}{4}, n=3. (b) ln(x)x2(x0=1,n=3)\frac{\ln (x)}{x^{2}}\left(x_{0}=1, n=3\right).
2. Determine Determine the limited development in the neighborhood of ++\infty to the indicated order nn of the following functions : (a) f(x)=x+2x,n=3f(x)=\frac{\sqrt{x+2}}{\sqrt{x}}, n=3 (b) ln(x+1+x2)lnx(n=4)\ln \left(x+\sqrt{1+x^{2}}\right)-\ln x(n=4)

Studdy Solution
Simplify the expression.

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