Dengan teorema L'Hospital hitunglah limit-limit fungsi berikut. x→0lim x22−2 cos 2x

Pertanyaan

Dengan teorema L'Hospital hitunglah limit-limit fungsi berikut.

$begin mathsize 14px style limit as x rightwards arrow 0 of space fraction numerator 2 minus 2 space cos space 2 x over denominator x squared end fraction end style$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

$begin mathsize 14px style limit as x rightwards arrow 0 of space fraction numerator 2 minus 2 space cos space 2 x over denominator x squared end fraction equals 4 end style$

Pembahasan

Limit fungsi trigonometri: $begin mathsize 14px style limit as x rightwards arrow 0 of space fraction numerator sin space a x over denominator a x end fraction equals 1 end style$

Dengan teorema L'Hospital, nilai limit fungsi tersebut dapat ditentukan sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 0 of space fraction numerator 2 minus 2 space cos space 2 x over denominator x squared end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator negative 2 space open parentheses negative 2 space sin space 2 x close parentheses over denominator 2 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator 4 space sin space 2 x over denominator 2 x end fraction end cell row blank equals cell 4 times 1 end cell row blank equals 4 end table end style$

Dengan demikian, $begin mathsize 14px style limit as x rightwards arrow 0 of space fraction numerator 2 minus 2 space cos space 2 x over denominator x squared end fraction equals 4 end style$