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Let f(x)=lim(m->oo){lim(n->oo)cos^(2m)(n...

Let `f(x)=lim_(m->oo){lim_(n->oo)cos^(2m)(n !pix)},` where `x in Rdot` Then prove that `f(x)={1,`if x is rational` and 0, `if x is irrational``

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CENGAGE ENGLISH-LIMITS AND DERIVATIVES-All Questions
  1. Evaluate: ("lim")(xvec1)sec(pi/(2^x))logxdot

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  2. Evaluate: ("lim")(xvec0)(sinx)/x,w h e r e[dot] represents the greate...

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  3. Let f(x)=lim(m->oo){lim(n->oo)cos^(2m)(n !pix)}, where x in Rdot Then...

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  4. Evaluate: lim(x->pi/2)(1+cos2x)/((pi-2x)^2)

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  5. Evaluate underset(ntooo)limn^(-n^(2))[(n+2^(0))(n+2^(-1))(n+2^(-2))......

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  6. Solve: lim {x->oo} 2^(x-1) tan(a/(2^x))

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  7. Evaluate underset(xto pi//2)lim(sinx-(sinx)^(sinx))/(1-sinx+log(e)sinx...

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  8. Evaluate: lim(x->2)(x^2-x-2)/(x^2-2x-"sin"(x-2))

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  9. Evaluate: lim(x->0)(1/(x^2)-1/(sin^2x))

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  10. Evaluate: lim(x->oo)x(tan^(-1)((x+1)/(x+4))-pi/4)

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  11. Evaluate the value of ("lim")(n vec pi/2 )tan^2xsqrt((2sin^2x+3sinx...

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  12. Evaluate the limit: ("lim")(xvec1)("sin"(logx))/(logx)

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  13. If the difference between the roots of the equation x^2+""a x""+""1...

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  14. Evaluate: ("lim")(xvec0)1/xsin^(-1)((2x)/(1+x^2))

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  15. At the endpoint and midpoint of a circular are AB, tangent lines are ...

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  16. Evaluate: lim(x->0) (1-cos2x)/(x^2)

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  17. Evaluate ("lim")(x to 0)(x-sinx)/(x^3)dot (Do not use either LHospita...

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  18. Evaluate: ("lim")(x to 0)(tanx-sinx)/(x^3)

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  19. The value of ("lim")(n to 0)[1/n+(e^(1/n))/n+(e^(2/n))/n++(e^((n-1)/n)...

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  20. Find the values of a and b in order that lim(x->0) (x(1+a cos x)-b sin...

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