Untuk memudahkan dalam menentukan nilai limit suatu fungsi, kita butuh yang namanya sifat-sifat limit fungsi. Sifat-sifat limit fungsi merupakan suatu teorema yang digunakan dalam menyelesaikan limit suatu fungsi. Untuk menyelesaikan limit suatu fungsi ada berbagai cara, salah satu adalah dengan substitusi yang akan kita gunakan pada artikel kali ini. Silahkan juga baca materi "pengertian limit fungsi".
Contoh :
1). Tentukan nilai limit dari bentuk berikut :
a). $ \displaystyle \lim_{x \to 2 } 2x + 1 $
b). $ \displaystyle \lim_{x \to -1 } \frac{x^2 + 2}{2x - 1 } $
Penyelesaian :
a). $ \displaystyle \lim_{x \to 2 } 2x + 1 = 2(2) + 1 = 4 + 1 = 5 $
artinya nilai $ \displaystyle \lim_{x \to 2 } 2x + 1 = 5 $
b). $ \displaystyle \lim_{x \to -1 } \frac{x^2 + 2}{2x - 1 } = \frac{(-1)^2 + 2}{2(-1) - 1 } = \frac{1 + 2 }{-2-1} = \frac{3}{-3} = -1 $
artinya nilai $ \displaystyle \lim_{x \to -1 } \frac{x^2 + 2}{2x - 1 } = -1 $
Contoh :
2). Tentukan nilai limit fungsi berikut dengan menggunakan sifat-sifat yang ada,
a). $ \displaystyle \lim_{x \to 2 } 5 $
b). $ \displaystyle \lim_{x \to 3 } 2x^3 $
c). $ \displaystyle \lim_{x \to 1 } x^2 + x $
d). $ \displaystyle \lim_{x \to -1 } x^2 - 3x $
e). $ \displaystyle \lim_{x \to -2 } x^3.x^2 $
f). $ \displaystyle \lim_{x \to 3 } \frac{x^2 - 1}{x + 1} $
g). $ \displaystyle \lim_{x \to 2 } (2x^2 + 3)^9 $
h). $ \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } $
Penyelesaian :
a). $ \displaystyle \lim_{x \to 2 } 5 = 5 $
b). $ \displaystyle \lim_{x \to 3 } 2x^3 = 2 . \displaystyle \lim_{x \to 3 } x^3 = 2. 3^3 = 2. 37 = 74 $
c). $ \displaystyle \lim_{x \to 1 } x^2 + x = ..... $
$ \begin{align} \displaystyle \lim_{x \to 1 } x^2 + x & = \displaystyle \lim_{x \to 1 } x^2 + \displaystyle \lim_{x \to 1 } x \\ & = 1^2 + 1 \\ & = 1 + 1 = 2 \end{align} $
d). $ \displaystyle \lim_{x \to -1 } x^2 - 3x = ..... $
$ \begin{align} \displaystyle \lim_{x \to -1 } x^2 - 3x & = \displaystyle \lim_{x \to -1 } x^2 - \displaystyle \lim_{x \to -1 } 3x \\ & = \displaystyle \lim_{x \to -1 } x^2 - 3.\displaystyle \lim_{x \to -1 } x \\ & = (-1)^2 - 3.(-1) \\ & = 1 + 3 = 4 \end{align} $
e). $ \displaystyle \lim_{x \to -2 } x^3.x^2 = ..... $
$ \begin{align} \displaystyle \lim_{x \to -2 } x^3.x^2 & = \displaystyle \lim_{x \to -2 } x^3 . \displaystyle \lim_{x \to -2 } x^2 \\ & = (-2)^3 . (-2)^2 \\ & = -8 . 4 = -32 \end{align} $
f). $ \displaystyle \lim_{x \to 3 } \frac{x^2 - 1}{x + 1} = ..... $
$ \begin{align} \displaystyle \lim_{x \to 3 } \frac{x^2 - 1}{x + 1} & = \frac{ \displaystyle \lim_{x \to 3 } x^2 - 1}{ \displaystyle \lim_{x \to 3 } x + 1} \\ & = \frac{ \displaystyle \lim_{x \to 3 } x^2 - \displaystyle \lim_{x \to 3 } 1}{ \displaystyle \lim_{x \to 3 } x + \displaystyle \lim_{x \to 3 } 1} \\ & = \frac{ 3^2 - 1 }{ 3 + 1 } \\ & = \frac{ 8 }{ 4 } = 2 \end{align} $
g). $ \displaystyle \lim_{x \to 2 } (2x^2 + 3)^9 = ..... $
$ \begin{align} \displaystyle \lim_{x \to 2 } (2x^2 + 3)^9 & = \left( \displaystyle \lim_{x \to 2 } 2x^2 + 3 \right)^9 \\ & = \left( \displaystyle \lim_{x \to 2 } 2x^2 + \displaystyle \lim_{x \to 2 } 3 \right)^9 \\ & = \left( 2. \displaystyle \lim_{x \to 2 } x^2 + \displaystyle \lim_{x \to 2 } 3 \right)^9 \\ & = \left( 2. 2^2 + 3 \right)^9 \\ & = \left( 8 + 3 \right)^9 \\ & = \left( 11 \right)^9 \end{align} $
h). $ \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } = ..... $
$ \begin{align} \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } & = \sqrt[3]{ \displaystyle \lim_{x \to 3 } x^2 - 1 } \\ & = \sqrt[3]{ \displaystyle \lim_{x \to 3 } x^2 - \displaystyle \lim_{x \to 3 } 1 } \\ & = \sqrt[3]{ 3^2 - 1 } \\ & = \sqrt[3]{ 8 } = 2 \end{align} $
Catatan : Untuk menyelesaikan limit, bisa langsung substitusi saja tanpa harus dipecah menggunakan sifat-sifat yang ada karena hasilnya juga sama.
Contoh :
$ \begin{align} \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } = \sqrt[3]{ 3^2 - 1 } = \sqrt[3]{ 8 } = 2 . \end{align} $
Menyelesaikan limit dengan cara substitusi
Cara substitusi maksudnya langsung nilai $ x \, $ kita substitusi ke fungsi $ f(x) $. Contohnya : $ \displaystyle \lim_{x \to a } f(x) = f(a) $
1). Tentukan nilai limit dari bentuk berikut :
a). $ \displaystyle \lim_{x \to 2 } 2x + 1 $
b). $ \displaystyle \lim_{x \to -1 } \frac{x^2 + 2}{2x - 1 } $
Penyelesaian :
a). $ \displaystyle \lim_{x \to 2 } 2x + 1 = 2(2) + 1 = 4 + 1 = 5 $
artinya nilai $ \displaystyle \lim_{x \to 2 } 2x + 1 = 5 $
b). $ \displaystyle \lim_{x \to -1 } \frac{x^2 + 2}{2x - 1 } = \frac{(-1)^2 + 2}{2(-1) - 1 } = \frac{1 + 2 }{-2-1} = \frac{3}{-3} = -1 $
artinya nilai $ \displaystyle \lim_{x \to -1 } \frac{x^2 + 2}{2x - 1 } = -1 $
Sifat-sifat Limit Fungsi
Berikut sifat-sifat limit fungsi :
i). $ \displaystyle \lim_{x \to a } k = k \, $ dengan $ k \, $ adalah konstanta.
ii). $ \displaystyle \lim_{x \to a } k f(x) = k \displaystyle \lim_{x \to a } f(x) $
iii). $ \displaystyle \lim_{x \to a } [f(x) \pm g(x) ] = \displaystyle \lim_{x \to a } f(x) \pm \displaystyle \lim_{x \to a } g(x) $
iv). $ \displaystyle \lim_{x \to a } [f(x). g(x)] = \left( \displaystyle \lim_{x \to a } f(x) \right) \left( \displaystyle \lim_{x \to a } g(x) \right) $
v). $ \displaystyle \lim_{x \to a } \frac{f(x)}{g(x)} = \frac{ \displaystyle \lim_{x \to a } f(x) }{\displaystyle \lim_{x \to a } g(x) } $
vi). $ \displaystyle \lim_{x \to a } [f(x)]^n = \left[ \displaystyle \lim_{x \to a } f(x) \right]^n $
vii). $ \displaystyle \lim_{x \to a } \sqrt[n]{f(x)} = \sqrt[n]{\displaystyle \lim_{x \to a } f(x) } $
i). $ \displaystyle \lim_{x \to a } k = k \, $ dengan $ k \, $ adalah konstanta.
ii). $ \displaystyle \lim_{x \to a } k f(x) = k \displaystyle \lim_{x \to a } f(x) $
iii). $ \displaystyle \lim_{x \to a } [f(x) \pm g(x) ] = \displaystyle \lim_{x \to a } f(x) \pm \displaystyle \lim_{x \to a } g(x) $
iv). $ \displaystyle \lim_{x \to a } [f(x). g(x)] = \left( \displaystyle \lim_{x \to a } f(x) \right) \left( \displaystyle \lim_{x \to a } g(x) \right) $
v). $ \displaystyle \lim_{x \to a } \frac{f(x)}{g(x)} = \frac{ \displaystyle \lim_{x \to a } f(x) }{\displaystyle \lim_{x \to a } g(x) } $
vi). $ \displaystyle \lim_{x \to a } [f(x)]^n = \left[ \displaystyle \lim_{x \to a } f(x) \right]^n $
vii). $ \displaystyle \lim_{x \to a } \sqrt[n]{f(x)} = \sqrt[n]{\displaystyle \lim_{x \to a } f(x) } $
2). Tentukan nilai limit fungsi berikut dengan menggunakan sifat-sifat yang ada,
a). $ \displaystyle \lim_{x \to 2 } 5 $
b). $ \displaystyle \lim_{x \to 3 } 2x^3 $
c). $ \displaystyle \lim_{x \to 1 } x^2 + x $
d). $ \displaystyle \lim_{x \to -1 } x^2 - 3x $
e). $ \displaystyle \lim_{x \to -2 } x^3.x^2 $
f). $ \displaystyle \lim_{x \to 3 } \frac{x^2 - 1}{x + 1} $
g). $ \displaystyle \lim_{x \to 2 } (2x^2 + 3)^9 $
h). $ \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } $
Penyelesaian :
a). $ \displaystyle \lim_{x \to 2 } 5 = 5 $
b). $ \displaystyle \lim_{x \to 3 } 2x^3 = 2 . \displaystyle \lim_{x \to 3 } x^3 = 2. 3^3 = 2. 37 = 74 $
c). $ \displaystyle \lim_{x \to 1 } x^2 + x = ..... $
$ \begin{align} \displaystyle \lim_{x \to 1 } x^2 + x & = \displaystyle \lim_{x \to 1 } x^2 + \displaystyle \lim_{x \to 1 } x \\ & = 1^2 + 1 \\ & = 1 + 1 = 2 \end{align} $
d). $ \displaystyle \lim_{x \to -1 } x^2 - 3x = ..... $
$ \begin{align} \displaystyle \lim_{x \to -1 } x^2 - 3x & = \displaystyle \lim_{x \to -1 } x^2 - \displaystyle \lim_{x \to -1 } 3x \\ & = \displaystyle \lim_{x \to -1 } x^2 - 3.\displaystyle \lim_{x \to -1 } x \\ & = (-1)^2 - 3.(-1) \\ & = 1 + 3 = 4 \end{align} $
e). $ \displaystyle \lim_{x \to -2 } x^3.x^2 = ..... $
$ \begin{align} \displaystyle \lim_{x \to -2 } x^3.x^2 & = \displaystyle \lim_{x \to -2 } x^3 . \displaystyle \lim_{x \to -2 } x^2 \\ & = (-2)^3 . (-2)^2 \\ & = -8 . 4 = -32 \end{align} $
f). $ \displaystyle \lim_{x \to 3 } \frac{x^2 - 1}{x + 1} = ..... $
$ \begin{align} \displaystyle \lim_{x \to 3 } \frac{x^2 - 1}{x + 1} & = \frac{ \displaystyle \lim_{x \to 3 } x^2 - 1}{ \displaystyle \lim_{x \to 3 } x + 1} \\ & = \frac{ \displaystyle \lim_{x \to 3 } x^2 - \displaystyle \lim_{x \to 3 } 1}{ \displaystyle \lim_{x \to 3 } x + \displaystyle \lim_{x \to 3 } 1} \\ & = \frac{ 3^2 - 1 }{ 3 + 1 } \\ & = \frac{ 8 }{ 4 } = 2 \end{align} $
g). $ \displaystyle \lim_{x \to 2 } (2x^2 + 3)^9 = ..... $
$ \begin{align} \displaystyle \lim_{x \to 2 } (2x^2 + 3)^9 & = \left( \displaystyle \lim_{x \to 2 } 2x^2 + 3 \right)^9 \\ & = \left( \displaystyle \lim_{x \to 2 } 2x^2 + \displaystyle \lim_{x \to 2 } 3 \right)^9 \\ & = \left( 2. \displaystyle \lim_{x \to 2 } x^2 + \displaystyle \lim_{x \to 2 } 3 \right)^9 \\ & = \left( 2. 2^2 + 3 \right)^9 \\ & = \left( 8 + 3 \right)^9 \\ & = \left( 11 \right)^9 \end{align} $
h). $ \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } = ..... $
$ \begin{align} \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } & = \sqrt[3]{ \displaystyle \lim_{x \to 3 } x^2 - 1 } \\ & = \sqrt[3]{ \displaystyle \lim_{x \to 3 } x^2 - \displaystyle \lim_{x \to 3 } 1 } \\ & = \sqrt[3]{ 3^2 - 1 } \\ & = \sqrt[3]{ 8 } = 2 \end{align} $
Catatan : Untuk menyelesaikan limit, bisa langsung substitusi saja tanpa harus dipecah menggunakan sifat-sifat yang ada karena hasilnya juga sama.
Contoh :
$ \begin{align} \displaystyle \lim_{x \to 3 } \sqrt[3]{ x^2 - 1 } = \sqrt[3]{ 3^2 - 1 } = \sqrt[3]{ 8 } = 2 . \end{align} $