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Find the sum of an infinite geometric se...

Find the sum of an infinite geometric series whose first term is the limit of the function `f( x) =( tan x - sinx )/( sin^(3) x ) ` as `x rarr 0 ` and whose common ratio is the limit of the function `g(x) = ( 1-sqrt(x))/( (cos^(-1)x )` as `x rarr 1 `

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The correct Answer is:
`a = ( 1)/( 2) ; r = ( 1)/( 4); S= ( 2)/( 3)`
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MOTION-LIMIT-EXERCISE-3
  1. Evaluate : ("lim")(xrarroo) 1/(1+n^2)+2/(2+n^2)+...+n/(n+n^2)

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  2. underset( x rarr (pi)/(4))("Lim")(1-tanx)/( 1- sqrt(2) sin x)

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  3. Find the sum of an infinite geometric series whose first term is the l...

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  4. Evaluate the following limit: (lim)(x->1/(sqrt(2)))(cos^(-1)(2xsqrt...

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  5. underset( x rarr ( pi )/( 4))("Lim")(sqrt( 1- sqrt( sin x)))/(pi - 4x)

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  6. underset( x rarr (3pi)/( 4))("Lim")(1+ root( 3)(tan x))/( 1-2 cos^(2) ...

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  7. Evaluate: ("lim")(nvec0)8/(x^8){1-cos(x^2)/2-cos(x^2)/4+cos(x^2)/2cos(...

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  8. If lim(xto-oo)((3x^(4)+2x^(2)).sin(1/x)+|x|^(3)+5)/(|x|^(3)+|x^(2)|+|x...

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  9. lim(n->oo)sum(r=2)^n((r+1)sinpi/(r+1)-rsinpi/r)

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  10. underset( xrarr oo )("Lim")x^(2) sin(lnsqrt( cos ""( pi)/( x)))

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  11. Letf(x) = underset( n rarr oo)("Lim")underset( n=1)overset( n ) ( sum)...

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  12. Let f(x) = underset(n rarr oo)("Lim")( 2x^(2n) sin""(1)/(x) +x)/(1+x^(...

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  13. Let f(x) = underset(n rarr oo)("Lim")( 2x^(2n) sin""(1)/(x) +x)/(1+x^(...

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  14. Let f(x) = underset(n rarr oo)("Lim")( 2x^(2n) sin""(1)/(x) +x)/(1+x^(...

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  15. Let f(x) = underset(n rarr oo)("Lim")( 2x^(2n) sin""(1)/(x) +x)/(1+x^(...

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  16. Evaluate the limit if exists, lim(x->1)[{ln(1+x)-ln2}{3.4^[x-1]-3x}]...

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  17. lim(x->0)(2 7^x-9^x-3^x+1)/(sqrt(2)-sqrt(1+cosx))

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  18. lim(x->0) tan^(-1) (a/x^2), where a in R

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  19. Plot the graph of the function f(x)=lim(t->0)((2x)/pitan^(- 1)x/(t^2))

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  20. lim(x->oo)((2x^2+3)/(2x^2+5))^(8x^2+3)

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