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Pengenalan Limit ~ Matematika SMA | Catatan Nurkholis

Pengenalan Limit ~ Matematika SMA

Pengertian

Misalkan suatu fungsi $f(x)$ didefinisikan di sekitar $x=a$, maka
$$lim_{xto a}f(x)=L$$
diartikan untuk $x$ mendekati $a$ mengakibatkan fungsi $f(x)$ mendekati $L$.

Contoh:

Jika $f(x)=2x+1$, tentukan nilai $displaystylelim_{xto0}f(x)$

Solusi:

Perhatikan tabel berikut!

$x$
$f(x)=2x+1$
$x$
$f(x)=2x+1$
1
3
-1
-1
0,1
1,2
-0,1
0,8
0,01
1,02
-0,01
0,98
0,001
1,002
-0,001
0,998
0,0001
1.0002
-0,0001
0,9998
0,0000000001
1,0000000002
-0,0000000001
0,9999999998

Kita tahu bahwa fungsi $f(x)=2x+1$ memiliki nilai yang mendekati $1$ ketika $x$ mendekati $0$, baik saat $x$ lebih dari $0$ maupun kurang dari $0$ (kita notasikan $x=0^+$ dan $x=0^-$.
Secara ringkas dapat kita tuliskan
$$begin{aligned}
lim_{xto0^+}2x+1&=1\
lim_{xto0^-}2x+1&=1
end{aligned}$$
Sehingga dapat kita simpulkan bahwa
$$lim_{xto0}2x+1=2(0)+1=1$$

Problem:

Berapakah nilai $displaystylelim_{xto0}frac{1}{x}$?

Solusi:


$x$
$f(x)=1/x$
$x$
$f(x)=1/x$
1
1
-1
-1
0,1
10
-0,1
-10
0,01
100
-0,01
-100
0,001
1000
-0,001
-1000
0,0001
10000
-0,0001
-10000
0,0000000001
10000000000
-0,0000000001
-10000000000

Perhatikan bahwa saat $x$ dari $1$ menuju $0$, nilai $f(x)=frac{1}{x}$ akan mendekati $infty$. Akan tetapi, apabila kita lihat pada saat $x$ berada di antara $-1$ dan $0$, didapat nilai $f(x)$ mendekati $-infty$.
Atau dapat kita tuliskan:
$$begin{aligned}
lim_{xto0^+}frac{1}{x}&=infty\
lim_{xto0^-}frac{1}{x}&=-infty
end{aligned}$$
Dengan demikian, bisa kita simpulkan bahwa
$$lim_{xto0}frac{1}{x}=text{tidak ada}$$

Latihan Soal:

Tentukanlah nilai limit pada fungsi berikut!

  1. $displaystylelim_{xto3}4x+9$
  2. $displaystylelim_{xto5}sqrt{x^2+10x-25}$
  3. $displaystylelim_{xtopi}sin2x$
  4. $displaystylelim_{xto2pi}tanleft(frac{x}{4}right)$

Selamat Belajar!

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