Question and Answers Forum

AllQuestion and Answers: Page 1139

Question Number 101981 Answers: 3 Comments: 0

lim_(x→0) (((tanx)/x))^(1/x^2 )

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\left(\frac{{tanx}}{{x}}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} \\ $$

Question Number 101977 Answers: 0 Comments: 0

Given 2 functions, f and g, n-times derivable within the open interval, R and verify the property f(x_0 )=f^((k)) (x_0 )=0 , g(x_0 )=g^((k)) (x_0 )=0 , ∀k∈{1,2,...,n−1} Show that lim_(x→x_0 ) ((f(x))/(g(x)))=((f^((n)) (x_0 ))/(g^((n)) (x_0 )))

$$\mathrm{Given}\:\mathrm{2}\:\mathrm{functions},\:\mathrm{f}\:\mathrm{and}\:\mathrm{g},\:\mathrm{n}-\mathrm{times}\:\mathrm{derivable}\:\mathrm{within} \\ $$$$\mathrm{the}\:\mathrm{open}\:\mathrm{interval},\:\mathbb{R}\:\mathrm{and}\:\mathrm{verify}\:\mathrm{the}\:\mathrm{property} \\ $$$$\mathrm{f}\left(\mathrm{x}_{\mathrm{0}} \right)=\mathrm{f}^{\left(\mathrm{k}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)=\mathrm{0}\:,\:\mathrm{g}\left(\mathrm{x}_{\mathrm{0}} \right)=\mathrm{g}^{\left(\mathrm{k}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)=\mathrm{0}\:,\:\forall\mathrm{k}\in\left\{\mathrm{1},\mathrm{2},...,\mathrm{n}−\mathrm{1}\right\} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\underset{\mathrm{x}\rightarrow\mathrm{x}_{\mathrm{0}} } {\mathrm{lim}}\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}=\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)}{\mathrm{g}^{\left(\mathrm{n}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)} \\ $$

Question Number 101974 Answers: 2 Comments: 1

Question Number 101971 Answers: 0 Comments: 2

Question Number 101970 Answers: 1 Comments: 0

∫_0 ^∞ ((Cos(ax))/(x^2 +b^2 )) dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{Cos}\left({ax}\right)}{{x}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 101961 Answers: 0 Comments: 4

Question Number 101959 Answers: 1 Comments: 0

Starting from y=(4/(√π))t^((3/2) ) ∫_0 ^∞ x^3 e^(−tx^2 ) dx find (π/8)=?

$$ \\ $$$$ \\ $$$${Starting}\:{from} \\ $$$$\:\:\:\:\:\:\:\:{y}=\frac{\mathrm{4}}{\sqrt{\pi}}{t}^{\frac{\mathrm{3}}{\mathrm{2}}\:} \int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{3}} {e}^{−{tx}^{\mathrm{2}} } {dx} \\ $$$${find}\:\:\:\frac{\pi}{\mathrm{8}}=? \\ $$

Question Number 101951 Answers: 0 Comments: 3

Question Number 101940 Answers: 1 Comments: 0

lim_(n→∞ ) (1/n^2 )(ne^((−1)/n^2 ) +ne^((−4)/n^2 ) +.....∞)

$$\underset{{n}\rightarrow\infty\:} {\mathrm{lim}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left({ne}^{\frac{−\mathrm{1}}{{n}^{\mathrm{2}} }} +{ne}^{\frac{−\mathrm{4}}{{n}^{\mathrm{2}} }} +.....\infty\right) \\ $$

Question Number 101937 Answers: 0 Comments: 0

Given a circle with the center at the point O and the radius of the length R.From a point A outside so that AO=2R,drawing two tangents AB and AC to the circle (B and C are the tangency points).Take a arbitrary point M on smaller arc BC (M differ from B and C) The tangent pass M cuts AB and AC at Pand Q respectively.The segments OP and OQ cuts BC at D and E respectively. i)Prove that PQ=2DE ii)Define the position of M such the area of the triangle ODE is smallest and expression it by R

Question Number 101921 Answers: 1 Comments: 1

Please, help i cannot solve problems such as: x^4 =1, x^5 =1, or x^n =1 n∈N...

$$\boldsymbol{{Please}},\:\boldsymbol{{help}}\:\:\boldsymbol{{i}}\:\:\boldsymbol{{cannot}}\:\boldsymbol{{solve}}\:\boldsymbol{{problems}} \\ $$$$\boldsymbol{{such}}\:\boldsymbol{{as}}:\:\boldsymbol{{x}}^{\mathrm{4}} =\mathrm{1},\:\boldsymbol{{x}}^{\mathrm{5}} =\mathrm{1},\:\boldsymbol{{or}}\:\boldsymbol{{x}}^{\boldsymbol{{n}}} =\mathrm{1}\:\boldsymbol{{n}}\in\mathbb{N}... \\ $$

Question Number 101891 Answers: 3 Comments: 0

if x a integer number , when divided 8 has remainder 5 and divided 5 has remainder 2. find x

$${if}\:{x}\:{a}\:{integer}\:{number}\:,\:{when}\:{divided}\:\mathrm{8} \\ $$$${has}\:{remainder}\:\mathrm{5}\:{and}\:{divided}\:\mathrm{5}\:{has}\:{remainder} \\ $$$$\mathrm{2}.\:{find}\:{x} \\ $$

Question Number 101888 Answers: 3 Comments: 1

If x an integer when divided 5 give remainder 2 and when divided 4 give remainder 3. find the value of x

$${If}\:{x}\:{an}\:{integer}\:{when}\:{divided} \\ $$$$\mathrm{5}\:{give}\:{remainder}\:\mathrm{2}\:{and}\:{when}\:{divided} \\ $$$$\mathrm{4}\:{give}\:{remainder}\:\mathrm{3}. \\ $$$${find}\:{the}\:{value}\:{of}\:{x} \\ $$

Question Number 102323 Answers: 2 Comments: 0

Solve this linear Equation (dy/dx) + y cos x = (1/2) sin x

$${Solve}\:{this}\:{linear}\:{Equation} \\ $$$$\frac{{dy}}{{dx}}\:+\:{y}\:{cos}\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:{sin}\:{x} \\ $$