= x→0 lim(2 cosx/cos2x- 1)* sinx/x = 1

(because x→0, 2cosx/cos2x=2, sinx/x= 1)

2. the square of 2.x→ 0lim (cosx-cos3x)/x)/x

= x→ 0lim (cosx-cos3x)/x 2 (Robida's law, simultaneous derivation of upper and lower)

=x→0lim(-sinx+3sin3x)/2x (using Robida's law again)

= x→0 lim(-cosx+3c os3x)/2 = 1

3.x→0lim[tan(2x+x cubic) ]/sin(x-x square)

= x→ 0 lim [tan (2x+x 3)]/sin (x-x 2) (by Robida's law).

=x→0lim[sec^2(2x+x^3)(2+3x^2)]/cos(x-x^2)( 1-2x)

= 2(x→0,sec^2(2x+x^3)= 1,cos(x-x^2)= 1)

4.x→ positive infinity lim x*sin(2/x)

=x→ positive infinity lim sin(2/x)/( 1/x)

= t(= 1/x)→0 lim 2 * sin(2t)/(2t)= 2

5.x→0lim(2arcsinx)/3x (according to Robida's Law)

=x→0lim(2/√( 1-x^2))/3

=2/3

6.x→0lim(tanx-sinx)/sinx cubic.

= x→ 0lim (tanx-sinx)/sin 3x (according to Robida's law)

=x→0lim(sec^2x-cosx)/3sin^2xcosx

= x→ 0lim (sec 2x-cosx)/3sin 2x (according to Robida's law)

= x→ 0 lim (2secxtanxsecx+sinx)/6sinxcosx (according to Robida's law)

=x→0lim(2tanx+sinx)/6sinx (according to Robida's law)

=x→0lim(2sec^2x+cosx)/6cosx

= 1/2