Question

Sa se calculeze:
$\lim_{x \to 0} ( \frac{x^8-sin^8x}{x^{10}} )$

Answer 1

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Answer 2

lim sinx/x=1 pentru x tinzand la 0 si deci lim (sinx/x)^n=1 ptr x tinzand la 0

(x^8-sin^8(x))/x^10=(x^4-sin^4(x))*(x^4+sin^4(x)) /x^10=(x^2-sin^2(x))*(x^2+sin^2(x))*(x^4+sin^4(x)) /x^10= (x-sin(x)*(x+sin(x))*(x^2+sin^2(x))*(x^4+sin^4(x)) /x^10= (x-sin(x)/x^3 * (x+sin(x))/x * (x^2+sin^2(x))/x^2 * (x^4+sin^4(x)) /x^4

lim (x-sin(x)/x^3=lim (1-cosx)/3x^2= lim sinx/ 3*2*x =1/6 ptr x tinzand la 0

lim (x+sin(x))/x lim (1+sinx/x)=2 ptr x tinzand la 0

lim (x^2+sin^2(x))/x^2=lim (1+(sinx/x)^2)=2 ptr x tinzand la 0

lim (x^4+sin^4(x)) /x^4=2

deci L=1/6*2*2*2=4/3