Diberikan x→alimf(x)=3 dan x→alimg(x)=−1. Hitunglah nilai setiap limit berikut. a. x→alimf2(x)+g2(x) b. x→alimf(x)+g(x)2f(x)−3g(x) c. x→alim3g(x)[f(x)+3] d. x→alim[f(x)−3]4

Pertanyaan

Diberikan $\lim_{x\rightarrow a}f(x)=3$ dan $\style{font-size:14px}{\lim_{x\rightarrow a}g(x)=-1}$ .

Hitunglah nilai setiap limit berikut.

a. $\style{font-size:14px}{\lim_{x\rightarrow a}\sqrt{f^2(x)+g^2(x)}}$

b. $begin mathsize 14px style limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction end style$

c. $\style{font-size:14px}{\lim_{x\rightarrow a}\sqrt[3]{g(x)}\left[f(x)+3\right]}$

d. $\style{font-size:14px}{\lim_{x\rightarrow a}\left[f(x)-3\right]^4}$

P. Anggrayni

Master Teacher

Jawaban terverifikasi

Jawaban

hasil dari $\lim_{x\rightarrow a}\left[f(x)-3\right]^4$ adalah $0$ .

Pembahasan

Ingat bahwa:

Limit memiliki sifat seperti di bawah ini.

1. $\lim_{x\rightarrow a}\left[f(x)+g(x)\right]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$

2. $\lim_{x\rightarrow a}\left[f(x)-g(x)\right]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}g(x)$

3. $\lim_{x\rightarrow a}\left[f(x)\times g(x)\right]=\lim_{x\rightarrow a}f(x)\times\lim_{x\rightarrow a}g(x)$

4. $\lim_{x\rightarrow a}kf(x)=k\times\lim_{x\rightarrow a}f(x)$

5. $limit as x rightwards arrow a of open square brackets fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction close square brackets equals fraction numerator limit as x rightwards arrow a of f left parenthesis x right parenthesis over denominator limit as x rightwards arrow a of g left parenthesis x right parenthesis end fraction$

6. $\lim_{x\rightarrow a}\left[f(x)\right]^n=\left[\lim_{x\rightarrow a}f(x)\right]^n$

7. $\lim_{x\rightarrow a}\sqrt{f(x)}=\sqrt{\lim_{x\rightarrow a}f(x)}$

8. $\lim_{x\rightarrow a}c=c,\;c\in\mathbb{R}$

Diberikan ${\lim_{x\rightarrow a}f(x)=3}\;\mathrm{dan}\;\lim_{x\rightarrow a}g(x)=-1$

a. $\lim_{x\rightarrow a}\sqrt{f^2(x)+g^2(x)}$

Berdasarkan sifat 1, 6, dan 7 diperoleh:

$\begin{array}{rcl}\lim_{x\rightarrow a}\sqrt{f^2(x)+g^2(x)}&=&\sqrt{\left[\lim_{x\rightarrow a}f(x)\right]^2+\left[\lim_{x\rightarrow a}g(x)\right]^2}\\&=&\sqrt{\left[3\right]^2+\left[-1\right]^2}\\&=&\sqrt{9+1}\\&=&\sqrt{10}\end{array}$

Dengan demikian, hasil dari $\lim_{x\rightarrow a}\sqrt{f^2(x)+g^2(x)}$ adalah $\sqrt{10}$ .

b. $limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction$

Berdasarkan sifat 1, 2, 4, dan 5 diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction end cell equals cell fraction numerator 2 limit as x rightwards arrow a of f left parenthesis x right parenthesis minus 3 limit as x rightwards arrow a of g left parenthesis x right parenthesis over denominator limit as x rightwards arrow a of f left parenthesis x right parenthesis plus limit as x rightwards arrow a of g left parenthesis x right parenthesis end fraction end cell row blank equals cell fraction numerator 2 open square brackets 3 close square brackets minus 3 open square brackets negative 1 close square brackets over denominator 3 plus open parentheses negative 1 close parentheses end fraction end cell row blank equals cell fraction numerator 6 plus 3 over denominator 2 end fraction end cell row blank equals cell 9 over 2 end cell end table$

Dengan demikian, hasil dari $limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction$ adalah $\frac92$ .

c. $\lim_{x\rightarrow a}\sqrt[3]{g(x)}\left[f(x)+3\right]$

Berdasarkan sifat 1, 3, 7, dan 8 diperoleh:

$\begin{array}{rcl}\lim_{x\rightarrow a}\sqrt[3]{g(x)}\left[f(x)+3\right]&=&{\sqrt[3]{\lim_{x\rightarrow a}g(x)}\times\left[\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}3\right]}\\&=&\sqrt[3]{-1}\times\left[3+3\right]\\&=&-1\times6\\&=&-6\end{array}$

Dengan demikian, hasil dari $\lim_{x\rightarrow a}\sqrt[3]{g(x)}\left[f(x)+3\right]$ adalah $-6$ .

d. $\lim_{x\rightarrow a}\left[f(x)-3\right]^4$

Berdasarkan sifat 1, 6, dan 8 diperoleh:

$\begin{array}{rcl}\lim_{x\rightarrow a}\left[f(x)-3\right]^4&=&\left[\lim_{x\rightarrow a}\left[f(x)-3\right]\right]^4\\&=&\left[\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}3\right]^4\\&=&\left[3-3\right]^4\\&=&\left[0\right]^4\\&=&0\end{array}$

Dengan demikian, hasil dari $\lim_{x\rightarrow a}\left[f(x)-3\right]^4$ adalah $0$ .