Mat minatterakhir hari ini tengah malamcuma buat yg bener bener bisa jawab

Mat minatterakhir hari ini tengah malamcuma buat yg bener bener bisa jawab

Penjelasan dengan langkah-langkah:

1. $lim_{x to frac{pi }{3} } frac{cosx+sin2x}{tanx}$

$lim_{x to frac{pi }{3} } frac{cosx+sin2x}{tanx}\=frac{lim_{x to frac{pi }{3} } (cosx+sin2x)}{lim_{x to frac{pi }{3} } tanx}\=frac{lim_{x to frac{pi }{3} } cosx+lim_{x to frac{pi }{3} }sin2x}{lim_{x to frac{pi }{3} } tanx}\=frac{cos(frac{pi }{3} )+sin2(frac{pi }{3} )}{tan(frac{pi }{3} )}\=frac{cos(frac{pi }{3} )+sin(frac{2pi }{3} )}{tan(frac{pi }{3} )}\=frac{frac{1}{2}+frac{1}{2}sqrt{3} }{sqrt{3} }$

Rasionalkan

$=frac{frac{1}{2}+frac{1}{2}sqrt{3} }{sqrt{3} }.frac{sqrt{3} }{sqrt{3} } \=frac{frac{1}{2} sqrt{3}(1+sqrt{3} )}{3}\=frac{sqrt{3}(1+sqrt{3} )}{6}$

2. $lim_{x to 0} frac{sin4x-sinx}{2sin3xcosx}$

$lim_{x to 0} frac{sin4x-sinx}{2sin3xcosx} \= lim_{x to 0} frac{2cosfrac{1}{2}(5x)sinfrac{1}{2}(3x) }{2sin3xcosx}\= lim_{x to 0}frac{cosfrac{5x}{2}sinfrac{3x}{2} }{sin3xcosx} \= lim_{x to 0}frac{sinfrac{3x}{2} }{sin3x}.lim_{x to 0}frac{cosfrac{5x}{2}}{cosx}\=frac{frac{3}{2} }{3}.frac{cos0}{cos0} \=frac{3}{6} .1\=frac{1}{2}$

3. $lim_{x to 0} frac{1-cos6x}{4x^2}$

$lim_{x to 0} frac{1-cos6x}{4x^2} \= lim_{x to 0} frac{1-(1-2sin^23x)}{4x^2} \= lim_{x to 0} frac{1-1+2sin^23x}{4x^2} \= lim_{x to 0} frac{2sin^23x}{4x^2} \= lim_{x to 0} frac{sin3x.sin3x}{2x^2} \=lim_{x to 0} frac{1}{2} lim_{x to 0} frac{sin3x}{x}.lim_{x to 0} frac{sin3x}{x}\=frac{1}{2} .3.3\=frac{9}{2}$

4. $lim_{x to 0} frac{cos5x-cosx}{cos4x-1}$

$lim_{x to 0} frac{cos5x-cosx}{cos4x-1}\=lim_{x to 0} frac{-2sin3xsin2x}{(1-2sin^2x)-1}\=lim_{x to 0} frac{-2sin3xsin2x}{-2sin^2x}\=lim_{x to 0} frac{sin3xsin2x}{sin2xsin2x}\=lim_{x to 0} frac{sin3x}{sin2x}.lim_{x to 0} frac{sin2x}{sin2x}\=frac{3}{2}.frac{2}{2} \=frac{3}{2}$