Pertanyaan

Nilai x → 2 lim x 2 − 2 x x 2 + 5 − 3 adalah ....

Nilai $x \to 2 lim \frac{x ^{2} + 5 - 3}{x ^{2} - 2 x}$ adalah ....

P. Afrisno

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

nilai .

nilai $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator square root of x squared plus 5 end root minus 3 over denominator x squared minus 2 x end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 third end cell end table$ .

Pembahasan

Jika limit dari suatu fungsi tidak bisa ditentukan dengan substitusi langsung maka dapat ditentukan dengan cara pemfaktoran atau mengalikar akar sekawan. Misalnya diberikan fungsi , maka fungsi tersebut dapat dikalikan dengan akar sekawan dari pembilangnya sebagai berikut. Sehingga limit dari fungsi yang diminta pada soal di atas, dapat ditentukan sebagai berikut. Dengan demikian, nilai .

Jika limit dari suatu fungsi tidak bisa ditentukan dengan substitusi langsung maka dapat ditentukan dengan cara pemfaktoran atau mengalikar akar sekawan. Misalnya diberikan fungsi $fraction numerator square root of a x squared plus b end root plus c over denominator x minus d end fraction$ , maka fungsi tersebut dapat dikalikan dengan akar sekawan dari pembilangnya sebagai berikut.

$fraction numerator square root of a x squared plus b end root plus c over denominator x minus d end fraction times fraction numerator square root of a x squared plus b end root minus c over denominator square root of a x squared plus b end root minus c end fraction$

Sehingga limit dari fungsi yang diminta pada soal di atas, dapat ditentukan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space fraction numerator square root of x squared plus 5 end root minus 3 over denominator x squared minus 2 x end fraction end cell equals cell limit as x rightwards arrow 2 of space fraction numerator square root of x squared plus 5 end root minus 3 over denominator x squared minus 2 x end fraction times fraction numerator square root of x squared plus 5 end root plus 3 over denominator square root of x squared plus 5 end root plus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator x squared plus 5 minus 9 over denominator open parentheses x squared minus 2 x close parentheses open parentheses square root of x squared plus 5 end root plus 3 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator x squared minus 4 over denominator x open parentheses x minus 2 close parentheses open parentheses square root of x squared plus 5 end root plus 3 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator x squared minus 2 squared over denominator x open parentheses x minus 2 close parentheses open parentheses square root of x squared plus 5 end root plus 3 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses x plus 2 close parentheses up diagonal strike open parentheses x minus 2 close parentheses end strike over denominator x up diagonal strike open parentheses x minus 2 close parentheses end strike open parentheses square root of x squared plus 5 end root plus 3 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses x plus 2 close parentheses over denominator x open parentheses square root of x squared plus 5 end root plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses 2 plus 2 close parentheses over denominator 2 open parentheses square root of 2 squared plus 5 end root plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 4 over denominator 2 open parentheses square root of 9 plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 4 over denominator 2 times 6 end fraction end cell row blank equals cell 1 third end cell end table$

Dengan demikian, nilai $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator square root of x squared plus 5 end root minus 3 over denominator x squared minus 2 x end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 third end cell end table$ .