Pertanyaan

Tentukan nilai limit fungsi berikut! b. x → 1 lim x − 3 x + 1 − 2 dengan memisalkan x = t 2 − 1

Tentukan nilai limit fungsi berikut!

b. $x \to 1 lim \frac{x + 1 - 2}{x - 3}$ dengan memisalkan $x = t^{2} - 1$

L. Marlina

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

Jawaban

nilai limit adalah .

nilai limit $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator square root of x plus 1 end root minus 2 over denominator x minus 3 end fraction end cell end table end style$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator size 14px 1 over denominator square root of size 14px 2 size 14px plus size 14px 2 end fraction end cell end table$ .

Pembahasan

Dengan memisalkan x = t 2 − 1 maka: lim x → 1 x − 3 x + 1 − 2 = = = = = = lim t 2 − 1 → 1 ( t 2 − 1 ) − 3 t 2 − 1 + 1 − 2 lim t 2 − 1 → 1 t 2 − 4 t − 2 lim t 2 − 1 → 1 t 2 − 4 t − 2 lim t 2 − 1 → 1 ( t − 2 ) ( t + 2 ) t − 2 lim t → 2 ( t + 2 ) 1 2 + 2 1 Dengan demikian, nilai limit adalah .

Dengan memisalkan $x = t^{2} - 1$ maka:

$lim_{x \to 1} \frac{x + 1 - 2}{x - 3} = = = = = = lim_{t^{2} - 1 \to 1} \frac{t ^{2} - 1 + 1 - 2}{( t ^{2} - 1 ) - 3} lim_{t^{2} - 1 \to 1} \frac{t - 2}{t ^{2} - 4} lim_{t^{2} - 1 \to 1} \frac{t - 2}{t ^{2} - 4} lim_{t^{2} - 1 \to 1} \frac{t - 2}{( t - 2 ) ( t + 2 )} lim_{t \to 2} \frac{1}{( t + 2 )} \frac{1}{2 + 2}$

Dengan demikian, nilai limit $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator square root of x plus 1 end root minus 2 over denominator x minus 3 end fraction end cell end table end style$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator size 14px 1 over denominator square root of size 14px 2 size 14px plus size 14px 2 end fraction end cell end table$ .