Pertanyaan

Find the value of the limit. a. x → 0 lim x x + 2 − 2

Find the value of the limit.

a. $x \to 0 lim \frac{x + 2 - 2}{x}$

L. Marlina

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

Jawaban

.

$limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals 1 fourth square root of 2$ .

Pembahasan

Karena bentuk limit tersebut merupakan bentuk limit tak tentu, maka kita gunakan kali akar sekawan. Seperti berikut ini, Jadi, .

$limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals fraction numerator square root of 0 plus 2 end root minus square root of 2 over denominator 0 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator square root of 2 minus square root of 2 over denominator 0 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 0 over 0$

Karena bentuk limit tersebut merupakan bentuk limit tak tentu, maka kita gunakan kali akar sekawan. Seperti berikut ini,

$limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction. fraction numerator square root of x plus 2 end root plus square root of 2 over denominator square root of x plus 2 end root plus square root of 2 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator open parentheses x plus 2 close parentheses minus 2 over denominator x open parentheses square root of x plus 2 end root plus square root of 2 close parentheses end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator up diagonal strike x over denominator up diagonal strike x open parentheses square root of x plus 2 end root plus square root of 2 close parentheses end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator square root of x plus 2 end root plus square root of 2 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator square root of 0 plus 2 end root plus square root of 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of 2 end fraction. fraction numerator square root of 2 over denominator square root of 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 fourth square root of 2$

Jadi, $limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals 1 fourth square root of 2$ .