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Tentukan! g. x→4lim​x2−16x2−4x​

Pertanyaan

Tentukan!

g. begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction end style 

D. Kamilia

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

diperoleh begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction equals 1 half end style.

Pembahasan

Bentuk begin mathsize 14px style limit as x rightwards arrow a of f open parentheses x close parentheses comma space a not equal to 0 end style dan begin mathsize 14px style f open parentheses x close parentheses end style merupakan fungsi rasional (pecahan), lakukan tes limit dengan substitusi nilai begin mathsize 14px style x equals a end style ke fungsi undefined, jika diperoleh hasil begin mathsize 14px style 0 over 0 end style, lakukan faktorisasi undefined atas faktor linearnya sehingga penyebab bentuk begin mathsize 14px style 0 over 0 end style, dapat dihilangkan.

Tes limit dengan substitusi nilai begin mathsize 14px style x equals a end style ke fungsi undefined, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction end cell equals cell fraction numerator open parentheses 4 close parentheses squared minus 4 open parentheses 4 close parentheses over denominator open parentheses 4 close parentheses squared minus 16 end fraction end cell row blank equals cell fraction numerator 16 minus 16 over denominator 16 minus 16 end fraction end cell row cell limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction end cell equals cell 0 over 0 end cell end table end style  

Diperoleh hasil begin mathsize 14px style 0 over 0 end style, sehingga dilakukan faktorisasi undefined, perhatikan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction end cell equals cell limit as x rightwards arrow 4 of fraction numerator x open parentheses x minus 4 close parentheses over denominator open parentheses x plus 4 close parentheses open parentheses x minus 4 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator x over denominator x plus 4 end fraction end cell row blank equals cell fraction numerator open parentheses 4 close parentheses over denominator open parentheses 4 close parentheses plus 4 end fraction end cell row blank equals cell 4 over 8 end cell row cell limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction end cell equals cell 1 half end cell end table end style  

Jadi, diperoleh begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator x squared minus 4 x over denominator x squared minus 16 end fraction equals 1 half end style.

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